a theory of two-dimensional linear recurring arrays

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-18, NO. 6, NOVEMBER 1972

A Theory of Two-Dimens ional L inear Recurring Arrays

775

TAMIYA NOMURA, HIROSHI MIYAKAWA, MEMBER, IEEE, HIDEKI IMAI, AND AKIRA FUKUDA, STUDENT MEMBER, IEEE

Abstrucr-In this paper, two-dimensional arrays of elements of an arbitrary finite field are examined, especially arrays having maximum- area matrices. W e first define two-dimensional linear recurring arrays. In order to study the characteristics of two-dimensional linear recurring arrays, we also define two-dimensional linear cyclic codes. A systematic method of constructing two-dimensional linear recurring arrays having maximum-area matrices is given using the theory of two-dimensional cyclic codes. These arrays, here called y/Garrays, may be said to be two-dimensional analogs of M-sequences. A yp-array of area N, x NY exists over GF(q) if and only if N,N, is equal to qN - 1 for some positive integer N. Many interesting characteristics of the y/?-array, such as the propert ies of its autocorrelation function and the propert ies of the characteristic arrays, are deduced and explained.

I. INTR~DLJOTI~N

T

HE CODES that coding theory has treated in the past have been restricted almost exclusively to codes of one

dimension, i.e., codes that can be expressed on only one axis such as the time axis. This restriction seems to be partly due to the fact that currently available devices are not suitable for processing multidimensional information.

Some types of codes that could be described as multidimensional arrays have been studied. Product codes, for example, are codes of this type. However, product codes cannot be called multidimensional codes in the sense of this paper since they have been introduced only in order to construct better one-dimensional codes.

Many useful one-dimensional codes have been constructed to date and are being employed in various applications. This fact suggests that there must be multidimensional codes having distinctive properties that are analogous to those of one-dimensional codes. It is clear that the transmis- sion of multidimensional information does not necessarily require a multidimensional code. However, multidimensional codes can play an important role in some applications. For example, a two-dimensional code conceived as a multidimensional analog of an ordinary M-sequence might be useful for detecting two-dimensional shifts of an array of symbols.

Reed and Stewart [l], Spann [2], and Gordon [3] have made studies of the two-dimensional arrays of a so-called perfect map. The concept of the arrays of a perfect map is

Manuscript received July 23, 1971. T. Nomura and A. Fukuda are with the Institute of Space and

Aeronautical Science, University of Tokyo, Tokyo, Japan. H. Miyakawa is with the Faculty of Engineering, University of

Tokyo, Tokyo, Japan. H. Imai is with the Faculty of Engineering, Yokohama National

University, Yokohama, Japan.

almost similar to that of arrays having maximum-area matrices, which are defined in Section II of this paper. Calabro and Wolf [9] have made studies on some two- dimensional arrays having interesting properties and have covered many prior studies of these problems. These studies were quite interesting, but seemed to be somewhat lacking in generality.

We first define two-dimensional linear recurring arrays of elements of an arbitrary finite field. We define also two- dimensional linear cyclic codes and utilize them to analyze the characteristics of two-dimensional linear recurring arrays. A systematic method of constructing two-dimensional arrays having maximum-area matrices is presented using the results obtained through this analysis. These arrays are called yp-arrays.

These yp-arrays have many interesting characteristics that can be regarded as two-dimensional analogs of M-sequence properties, but they also have some other interesting characteristics that are not present in M-sequences. Two- dimensional linear recurring arrays and two-dimensional linear cyclic codes include the usual one-dimensional linear recurring sequences and linear cyclic codes.

II. TWO-DIMENSIONAL LINEAR RECURRING ARRAYS AND TWO-DIMENSIONAL LINEAR CYCLIC CODES

Two-Dimensional Linear Recurring Arrays

An infinite matrix A, over GF(q) (q is an arbitrary power of an arbitrary prime number)

[a,,, aO,l a0,2 + * * 1

A, = al,1

. . 1

3 aij E GF(q) (1)

is called here a two-dimensional array over GE’(q). If for arbitrary nonnegative integers I and J, the elements {aI+l,J+k I k = mandO I I< nor1 = nandO 5 k < m} of A, can be obtained from an n x m submatrix of A,

LaI+,,-l,, . . . aI+n-l,J+m-l J by applying the linear relations

n-l m-l

aI+l,J+k = z. j&o ci,j(“k)aI+i,J+j,

C. .(“k) E GF(q); k = m, 0 5 1 < n; 1 = n, 0 s k < m l,J (3)

776 IEEE TRANSACTIONS ON INFORMATION THEORY, NOVEMBER 1972

then A, is said to be a two-dimensional linear recurring array. The set of relations (3) is defined to be a two-dimensional linear recurring relation of degree (n,m). The array A, can be generated by this linear recurring relation, if an n x m matrix that is located at the position (0,O) is given initially.

Equations (3) do not always represent a linear recurring relation if an arbitrary set {Ci,j(‘Sk)} of the elements of GF(q) is used as the coefficients of (3), because there are many ways of determining an element ai,j by (3), and these ways must not contradict each other. A necessary and sufficient condition for the set {Ci,j(z*k)} to define a linear recurring relation without contradiction has already been obtained [4].

It is obvious that all n + m relations of (3) can be obtained from at most [min (n,m) + l] relations: n + 1 relations determining {al+l,J+k 1 k = m and 0 I I < n or k = 0 and I = n} for n 5 m, or m + 1 relations determining {aI+l,J+k 1 k = m and 1 = 0 or 0 I k < m and I = n} for n 2 m. These [min (n,m) + l] relations may be conveniently used for the purpose of constructing linear recurring arrays.

If there exist positive integers pX and pY satisfying

ai,j = ai+px,j = ai,j+p, (4)

for all i,j 2 0, then the period of A, is denoted by (p,.,p,,), where pX and pY are the least positive integers satisfying (4).

An array A, is said to have maximum-area matrices of area N, x NY if, for some positive integers n, and nY, all n, x ny submatrices are different from each other and every nonzero n, x nY matrix over GF(q) appears in an arbitrary Wx + n, - 1) x (NY + nY - 1) matrix contained in A,.

It is obvious that for an N, x NY matrix to be a maximum-area matrix a necessary condition is NXNY = 4 . “X”Y - 1

In this paper, an array having a period (p,,p,) and maximum-area matrices of area N, x NY is said to be an M-array if pX = N, and p,, = NY.

Two-Dimensional Linear Cyclic Codes

An N x M matrix over GF(q)

A = {ai,j} = aN-l,0 **. aN-l,M-l I

ai,j E GF(q) -

(5)

is called an N x M-array. The set V, X M of all these matrices is an NM-dimensional vector space over GE’(q). Every subspace C of V, X M is called a two-dimensional linear code of area N x M.

The three-dimensional matrix G, which is obtained by piling up N x M matrices G,,G,; . .,G, is called a generator matrix of the linear code C, where K is the dimension of the subspace C and {Gi} is a basis of C. Hereafter, G is expressed as G = {gi,j} where gi,j are the elements of the K-dimensional vector space. This means that G is regarded as an N x M matrix over the K-dimensional vector space.

The dual code of C is defined as a two-dimensional linear code that is a null space of C. (The inner product is defined by

N-l M-l

(A&) = C C ai,jbi,j, i=O j=O

where A = {ai,j} and B = {bi,j} are two elements of V, xM.) The generator matrix H = {hi,j} of this dual code is called a parity-check matrix of C, where the hi,j are elements of (NM - K)-dimensional vector space. It is evident that a necessary and sufficient condition for an N x M-array A = {ai,j} to be a codeword of C is

N-l M-l

(6)

A two-dimensional linear code C is said to be a two- dimensional linear cyclic code, if and only if any cyclic shift of any codeword A = {ai,j} is also a codeword of C. The cyclic shift of A = {ai,j} is an N x M-array {ai+k,j+l}, where k and 1 are arbitrary integers and the subscripts u and v of au,” are calculated modulo N and M, respectively.

It is convenient when studying the characteristics of two- dimensional linear cyclic codes to introduce a ring K[x,y], which consists of all two-variable polynomials over GF(q). A residue class ring of K[x,y] with respect to an ideal (x” - 1, yM - 1) is denoted by A(xN - 1, yM - 1). There exists a unique polynomial of degree less than N with respect to x and less than M with respect to y in every residue class of A(xN - 1, yM - 1). This can be shown in a way similar to that for the one-dimensional case [4]. A residue class will be represented by such a unique polynomial hereafter. The ring A(xN - 1, yM - 1) is an NM- dimensional vector space over GF(q) and is isomorphic to V N xM with respect to the correspondence a(x,y) c-) {ai,j}, where ai,j is the coefficient of the term x’yj of a(x,y). In this paper, polynomials are always regarded as the elements of A(2 - 1, yM - 1) if not otherwise indicated. Further- more, a polynomial

N-l M-l

a(x,y) = C C ai,jXiYj i=O j=O

is also considered to represent a two-dimensional array A = {ai,j}.

A two-dimensional linear code C is a two-dimensional linear cyclic code if and only if a(x, y) E C implies xky’a(x, y) E C, where k and I are arbitrary positive integers. It is easy to prove that a subspace C of the vector space A(xN - 1, YM - 1) is a two-dimensional linear cyclic code if and only if C is an ideal of the residue class ring A(xN - 1, yM - 1) [41.

Now, an infinite two-dimensional array of a finite array A can be regarded as an array A,. It is evident that such an array A has a period. Hereafter such an array A, is also, for brevity, denoted by A or a(x,y). The array A(= A,) can always be generated by means of some linear recurring relation if A is a codeword of a two-dimensional linear cyclic code. For example, parity-check polynomials of C [cf. (6)] can be used as the linear recurring relation.

NOMURA f?t a[. : TWO-DIMENSIONAL LINEAR ARRAYS

Examples: Let q = 2, n = 3, and m = 2. i) The array shown in Fig. 1 is obtained from the

linear recurring relation

af+3,5 = aI,, + a1t2,J

aI,Jt2 = aI,.I + aI,Jtl + aIt2,Jt1. (7)

This array is an M-array. ii) In Fig. 2, the array obtained from the linear recurring

relation (8) is shown.

aIt3,J = aI,J + aIt2,J

aI,Jt2 = aI,Jtl f aItl,J + aI+2,Jtl. (8) This array has maximum-area matrices but is not an M- array.

iii) From the linear recurring relation (9) the array that is shown in Fig. 3 is obtained. This array has no maximum- area matrices.

aIt3,J = aI,Jtl + aI+l,J + aIt2,J

aIt3,Jt1 = aI,J + aI,Jtl

aI,Jt2 = aI,J + aI,J+l + aItl,Jtl + aIt2,J+1. (9)

111. @ARRAYS

Dejinitions

Let n, m, p, and J. be positive integers that satisfy the following three conditions :

dlq” - 1 WV

if4” - ’ qk - 1 thenk 2 n (11) P

nm

gcd -1 A, 4 q” - 1 (12)

where q is a power of an arbitrary prime number and a/b denotes that b is divisible by a. Then, y and p are defined by

y = &z”“- l)/(q”- 1)lP p = CPA, (13)

where c( is a primitive element of GF(q”m) and 11 is a positive integer relatively prime to q”” - 1. It is obvious that y is an element of GF(q”).

It is easy to show that the orders of y and j? are given respectively by

e -q”-1 4 nm -1

Y eS = ;1 (14)

P

Let h,(x) be a minimal polynomial of y over GF(q) and /Z&J), a minimal polynomial of p over GF(q”). Then h,(x) and As(y), whose degrees are given by Lemma 1, are unique except for multiplication by constants, i.e., multiplication by nonzero elements of GF(q) and GF(q”), respectively [5].

Lemma 1: The minimal polynomials h,(x) and h@(y) are irreducible polynomials of degrees n and m, respectively.

Proof: The degree of h,(x) is given by the least positive integer d satisfying p(qd - 1) = 0 mod (q” - 1) [5]. It follows directly from (11) that d = n.

x

Fig. 1. Example of an M-array (A is a maximum-area matrix).

110010010 110111011 001101010 111111000 001000011

Fig. 2. Example of an array having maximum-area matrices (A is a maximum-area matrix).

x

V

0011110 0011300 0110101 7 00l0001 0010100 1101111

Fig. 3. Example of an array having no maximum-area matrices.

The degree of h&) is given by the least positive integer d satisfying $[(q”)d - l] = 0 mod ((9”)” - 1) [5]. This congruence is equivalent to

A(qnd - 1) = 0 mod (q”” - l),

because gcd (q, q”” - 1) = 1.

(15)

Clearly A < q” since it follows immediately from (10) that 1 1 q” - 1. Therefore, A(qnd - 1) I Aqnd - 1 < q”” - 1 if we suppose that d I m - 1. Then from (15), A(q”d - 1) = 0, which is a contradiction. Consequently, d must be greater than or equal to m. Then, the degree of h&) is m since d = m satisfies (15). Q.E.D.

Let P,-l = {a(x)} be a set of all polynomials over GF(q) of degrees less than n, and let a(x) E P,,- 1 correspond to a(y) E GF(q”). This correspondence between the elements of P,, _ I and the elements of GF(q”) is a one-to-one correspondence because h,(x) is an irreducible polynomial of degree n. Applying this correspondence, let the coefficients ci E GF(q”) of

correspond to ci(x) E P,- I and let m

,c- ci(x)Y i &=”


be represented by hYp(x,y). Notice here that h,&,y) = h,(y). Then a set of all elements f(x,y) of A(xey - 1, Y ea - 1) satisfying the following simultaneous equations (16) and (17) in A(xey - 1, yes - 1) is called here a yp-array code and is represented by C,,:

h,Wf(XYY) = 0 (16)

&?(&Y)f(X,Y) = 0. (17)

Every nonzero codeword of C,, is called a $-array.

Structures of a y/?-Array

It is evident from the definition that Cyp is a two-dimensional linear cyclic code of area eY x eS. Iff(x, y) is expressed as

then from (16) we have

0 I i < ea.

Consequently the fi(x) are codewords of a conventional one-dimensional cyclic code C, over GF(q) whose parity- check polynomial is h,(x).

Let a(x) E C,; then a(y) E GF(q”) and the correspondence a(x) e a(y) is a one-to-one correspondence. [If a(x), b(x) E C,, and a(x) = b(x) then a(y) = b(y). Conversely if a(y) = b(y) then g(x) = a(x) - b(x) has all e,th roots of unity as its roots. Therefore a(x) = b(x).] Accordingly, the correspondence f(x,y) of(y,y) is a one-to-one correspondence, where f(x, y) are the elements of A(xey - 1, Y ep - 1) that satisfy (16) and f(y, y) are the one-variable polynomials over GF(q”).

We can easily show that the correspondence between the solutions f(x, y) of (16), (17) and the solutions f(y, y) of

hp(~>f(r,~> = 0 mod (Y”” - 1) (19)

is a one-to-one correspondence, taking notice of the fact that the correspondences P,- I o GF(q”) and C, c> GF(q”) mentioned previously are both isomorphisms. In particular f(x,y) = 0 corresponds tof(y, y) = 0.

The congruence (19) has q”” solutions, and therefore CYs also has q”” codewords, because the degree of h,( y) is m.

Lemma 2: A necessary and sufficient condition for a polynomial a(x, y) to satisfy a(x, y)f(x, y) = 0 is a(y$) = 0, wheref(x,y) is a y/3-array.

Proof: From Lemma 2, a necessary and sufficient condition for the validity of (22) is

yipj = 1. (24)

Therefore, /I’ = ymi. Taking the e,th power of both terms, (fl’y)j = 1. Thus j must be divisible by the order of /I’?, which is denoted by ePr. Obviously epY is given by

while

Proof: From (19) f(y,P) # 0. (If f(yJ3) = 0 then f(y,y) has all epth roots of unity as its roots. Therefore f(y,y) must be zero and then f(x, y) = 0.) Consequently, if a(x,y)f(x,y) = 0 then a(y,P) = 0.

Let us regard hrp(x,y) as a polynomial in a variable y and let the coefficient of the term y” be c(x). There exists a polynomial C(x), c(x)Z(x) = 1 mod h,(x) because c(x) is a nonzero polynomial of degree less than n. Then the coefficient of the term y”’ of Z(x)hY8(x,y) takes the form 1 + k(x)h,(x). Consequently, any polynomial a(x,y) is divisible by a polynomial {i;(x)hYB(x,y) - k(x)h,(x)y”}

c, ~ -1 q”“-1 = 4”s gcd

nm gcd 4 -1

P 2 4 1,

q” - 1 /I

because

and

Therefore,

when they are both regarded as polynomials in y. Therefore,

a(x,y) can be decomposed as

&A = h,&,y)al(w) + hy(x)a2(x,Y) + a3(xA (20)

where the degree of a3(x, y) with respect to y is less than m. If a(y$) = 0 then a3(y,/?) = 0 from (20). Consequently, a,(y, y) must be zero. Therefore a,(x, y) can be decomposed as a3(x, y) = h,(x)a,(x, y). Thus

4w) = hy,kw)al(x,A + h,(xHa2(x,A + a&w>>.

From (16) and (17), a(x,y)f(x,y) = 0. Q.E.D.

From this lemma it follows directly that the least positive integers i,j satisfying

a-(X,Y) = f(X?Y)

Y-Y&Y) = fG%Y)

are eY and ea, respectively, wheref(x,y) is a y/?-array. This means that the period (p,,p,) of a yp-array is

(p,,p,) = (“,‘) F) . (21)

The following lemma is very important for the purpose of detailed discussion of the structures of yfl-arrays.

Lemma 3: Let f(x, y) be a y/?-array; then

x’y’f(x, y) = f(x, y), 0 I i < px, 0 I j < py (22)

if and only if

i = -$k mod pX, nm

j = 4 pk, q” - 1

0 I k < f$. (23)

e/J 4 nm -1

eSy = gcd (e,,e& = J. gcd [(q” - 1)/p, (4’“’ - O/4 ’

- 4” - l PA

gcd

pA 1 q” - 1.

eSr = (4”” - lMqn - 1).

NOMURA etal.: TWO-DIMENSIONAL LINEAR ARRAYS 119

Let j = ke,,. Then from (13) yipj = yipkepy = Yi+nlke

Therefore (24) is valid if and only if

i = -qAk mod e,,

j = keSv mod eB, k = O,l;...

of CYs is given by (28). Finally, we can express C,, as in (27), using an arbitrary nonzero codeword f (x, y) of C,,.

The structure of a y/I-array can be discussed in more detail by investigating the generator matrix of C,,.

Lemma 6: Let S be a subset having nm elements of the set of all ordered pairs of two integers i and j (0 I i < pX,

(25) 0 I j < p,). Suppose the nm elements {gi,j 1 (i,j) E S} of

Equation (23) can be derived directly from (25). Q.E.D. the generator matrix G = {gi,j} of C,, are linearly independent over GF(q). Let f (k*r)(x, y) be defined as

Now, N, and NY are defined by Px-1 PY-1

N, = ey = px = 4”-1 N, = esy = q”“- p. (26) f’“~‘Yx, y> = Jo jJlo fi,j(k’z)XiYj = xkY!f(x,Yh (29)

P qn - 1

CoroZZary 2 : Let f ‘(x, y) be the part of a y/?-array f (x, y) where f (x, y) is an arbitrary nonzero codeword of C,,. Let

that contains all terms of.f(x,y) whose degrees are less than wkl be the following ordered sets:

NY with respect to y. Then f (ii y) can be expressed as K-l

fc?Y) = ,& Xz(k)yNykf'(X, y),

where Z(k) = -r$k mod pX and K = (q” - l)/pA. From Lemma 3, x’yjf (x,y) = f (x,y) if and only if

i = j = 0 when i and j are restricted to 0 I i < N, and 0 < j < NY. Furthermore N,N, = q”” - 1 and there are 4 nm codewords in C,,. Thus we have proved the following lemma.

Lemma 4: C,, can be expressed using its nonzero codeword f (x, y) as

w kz = {fi,i(k*‘) I (i,j) E S}.

Then in the set {wkl 10 I k < N,., 0 I j < N,,} every ordered set of nm elements of GF(q), except for the set that contains no nonzero element, appears once and only once.

Proof: Let us define the ordered set w, for a(x, y)(E C,,) by O, = {ai,j 1 (i,j) E S}. Then in the set {ma 1 a(x,y) E C,,} every ordered set of nm elements of GF(q) appears once and only once, since the dimension of C,, is nm and the nm elements {gij 1 (i,j) E S} of G are linearly independent over GF(q). From this result and Lemma 4, this lemma follows directly. Q.E.D.

cyfl = {xiYif(x7Y) 1 0 5 i < N,, 0 5 .i < NJ U (0). (27) The Principaz Theorem

Now, let us derive the generator matrix of C,,. Let e,, be a set of all a”(~, y) such that

The equation

Li(x,y) = xPxyPra(x-‘,y-‘),

where a(x, y) is a polynomial that satisfies a(x, y)f (x, y) = 0 for a y/3-array f(x,y). It can easily be shown by methods similar to those of one-dimensional coding theory that & is the dual code of C,,. From Lemma 2,

px-1 PY-1

belongs to cYs if and only if b(y-‘J-l) = 0, i.e.,

n-l m-l n-l m-l C C &,jgi,j = C C ~i,jy-'B-' = 0, li,j E GF(q)

i=C) j=O i=O j=O

is valid if and only if all Ai,j are 0, because the degree of a minimal polynomial of /3 - ’ over GF(q”) is m and the degree of a minimal polynomial of y-l over GF(q) is n. That is, the nm elements {gi,j 1 i = 0, 1, * * *, n - 1 and j = 0, 1,. . . , m - 1 } of G are linearly independent over GF(q). From this fact and Lemma 6, we can obtain the most important theorem in this paper by setting n, = n and n,, = m in the definition of arrays having maximum-area matrices.

Theorem 7: A y/?-array has maximum-area matrices of area

Consequently, the parity-check matrix of c,,, i.e., the generator matrix of CYs, is expressed as N, x N, = (52) x (5 p) .

G = (gi,j> = (y-ip-j> = {CI-%i-q~j}, (28)

where G is a pX x pY matrix over GF(q”“). The principal results thus obtained are summarized in

the following theorem. Theorem 5: The y/?-array code C,, over GF(q), which is

defined by positive integers n,m,p,A satisfying (lo), (1 l), and (12), by a positive integer q that is relatively prime to 4 nm - 1, and by a primitive element CL of GF(q”“), is a two-dimensional linear cyclic code having q”” codewords. A codeword of C,, is a pX x p,, array. A generator matrix

In particular, if ~Lll = q” - 1, then the y/?-array is an M-array. This array is called a y@ M-array.

The following corollary is evident from the definition of the maximum-area matrix and Corollary 1.

Corollary 2: In an arbitrary N, x N,, array that is included in a y/I-array f (x, y), 0 appears q”‘“-I - 1 times and any other element of GF(q) appears qnmml times. In a period of f(x,y), 0 appears (qnmml - l)(q” - 1)/p,? times and any other element of GF(q) appears qnm-l(qn - l)/pA times.

180 IEEE TRANSACTIONS ON INFORMATION THEORY. NOVEMBER 1972

The Linear Recurring Relations Generating the y/l-Arrays

The y/?-array can be generated also by a linear recurring relation, which is derived from (16) and (17).

Select minimal polynomials h?(x) and hS(y) whose constant terms are 1 and y”-l, respectively. That is, h,(O) = 1 and h,(O) = y”-‘. Thus, IzYa(x,O) = x”-i. Then hi is defined as the coefficient of the term xi of h”,(x) = x”h,(x-‘) and hij is defined as the coefficient of the term xiyj of !&(x,y) = x”-lymhyS(x-l,y-l). Equations (16) and (17) can be rewritten as

n-1

f k+n,l = -igo &fk+i,l

n-l m-l

f k,l+m = - igo jgo ‘i,jfk+i,l+i~

where f,,, is the coefficient of the term x”y” of the yg-array f(x,y) and the subscripts u and u are calculated modulo pX and p,,, respectively.

Using (30) and (31), all fi,j can be determined succes- sively if initial values {fi,j 1 i = 0, 1,. . . , n - 1 and j = 0, l;**, m - l} are given.

Example: Let q = 2, H = 3, m = 2, p = 1, 2 = q” - 1 = 7, q = 1, and ~1~ + c1 + 1 = 0. The yfi-array defined by, those parameters is an M-array since ~1 = q” - 1. Further pX = N, = (q” - 1)/p = 7, p,, = N,, = (q”” - l)/ ,? = 9. The minimal polynomial of y = ~1~ over GF(2) whose constant term is 1 is by(x) = x3 + x2 + 1 and the minimal polynomial of /I = a7 over GF(23) whose constant term is yz is h,(y) = y2(y2 + (y + 1)y + 1). Thus hYB(x,y) = x2y2 + y + x2. Consequently h”,(x) = x3 + x + 1 and /&(x,y) = y2 + x2y + 1.

Equations (30) and (31) are expressed as

f k+3,1 = fk,l + fk+l,l

Y 110010103 111010101 111100011 001000001

D- 000110110 110100010 001110111

Fig. 4. Example of a yg-M-array (q = 2, n = 3, m = p = 1=7,q=l,a6+a+l=O).

2, 1,

(qN - 1)/K) = 1. At least one such A exists. For example, let 1 be equal to 1. Then it can be shown easily that these n, m, ~1, and 2 satisfy (IO), (1 l), and (12).

Let q be an arbitrary positive integer relatively prime to qN - 1 and let c1 be an arbitrary primitive element of GF(qN); then the $-array constructed using these parameters has maximum-area matrices of area K x L.

If K is relatively prime to L, then A can be equated to K. Then, FA = q” - 1, and in this case the y/?-array previously described is a yP M-array.

A Correspondence Between y/3-Arrays and M-Sequences

The minimal polynomial of CL over GF(q) is denoted by h,(z). The cyclic code C, of length q”” - 1 that is defined by the parity-check polynomial h,(z) is called the M- sequence code [S] or the shortened first-order Reed-Muller code [6]. The following theorem indicates a correspondence between the cyclic code C, and the y/I-array code C,,.

Theorem 9: Let Pe,- 1 be the set of all polynomials a(z) over GF(q) whose degrees are less than e,, where e, = 4 nm - 1, and let P be the set of all polynomials a’(x, y) over GF(q) whose degrees are less than N, with respect to x and less than N,, with respect to y. Furthermore, let Q be the set of all polynomials a(x,y) defined by the following equation :

f f k,Z+2 = k,l + fk+2,1+1* K-l

From this linear recurring relation the y/I M-array of @,y) = C x’(k)yNyka’(x, y),

k=O

Fig. 4 is obtained. l(k) = -qAk mod pX, IC = (4” - l)/pJ.

y/?-Arrays Having Maximum-Area Matrices of Area K x (qN - 1)/K Now, let 8 be a mapping from Q in P,=- 1 defined by

If a yfi-array over GF(q) is constructed, there must be a positive integer N that satisfies the relation N,N, = qN - 1. Furthermore, if the y/?-array is a y/I M-array, N, must be relatively prime to NY, since N, is equal to ;1 in that case.

Now, as a converse of the results so far obtained the next theorem can be derived.

Theorem 8: A y/?-array having maximum-area matrices of area K x L exists if K is an arbitrary divisor of qN - I and L = (qN - 1)/K for some positive integer N. Further- more a y/l M-array having maximum-area matrices of area K x L exists if in addition to the previously mentioned conditions K is relatively prime to L.

Proof: Let n be the least positive integer satisfying the conditions K 1 q” - I and n 1 N. It is obvious that such an integer n always exists. Then let p be defined by p = (q” - 1)/K and let k be a divisor of K where gcd (A,

8: a(x,y) + a(z) = a’(zNY,z4’) mod (z+ - l),

where a’(x, y) E P is the part of a(x, y) whose degree with respect to y is less than N,,.

Then 8 is a one-to-one mapping from Q onto P,.- 1. The y/3-array code C,, is contained in Q and the image of CYs is an M-sequence code C,. Conversely, the inverse image of any M-sequence code of length e, that is included in P,.- I is a @array code.

Proof: It is obvious that 8 is a composition mapping B, 0 B,, where

e1 : 4x, Y> + a’(x,y>

8,: a’(x,y) + a(z) = a’(zNY,zqrl) mod (zea - 1).

Obviously 8, is a one-to-one mapping onto P. On the other

NOMURA et cd. : TWO-DIMENSIONAL LINEAR ARRAYS

hand, 8, transfers the coefficient of the term x’y’ (0 < i < N,, 0 I j < NJ of a’(x,y) to the coefficient of the term z’ (I = NJ + $j mod e,) of a(z). Now let

I, = NJ1 + $jj, mod e,, 0 5 i,, i2 < N,;

I2 = Nyi2 + $j2 mod e,, 0 I jl, j, -c N,.

IfI, = I,, uZ~ = u12s Thus yilpjl = yi2pj2, i .e., yil-i2pjl-j2 = 1. Meanwhile, from the Proof of Lemma 3 (25) this is valid if and only if il = i, and j, = j, when i,, jl, i,, and j, are restricted by -N, < i, - i, < N,, -NY < j, - j, < NY. Furthermore N,N, = q”” - 1 = e,. Consequently e2 is a one-to-one map from (i,j) (0 I i < N,, 0 I j < NY) to 1 (0 I I < e,). This means that the coefficients of a(z) are obtained by the rearrangement of the coefficients of a’(x,y). Thus it is shown that 0 = e2 o 8, is a one-to one mapping from Q onto P,=- 1.

From Corollary 1 the codeword f (x, y) of C,, is contained in Q. The element of the generator matrix G of C,, that corresponds to the coefficient fi,j of f (x, y) is a-Nyi-tlLj = 6’ and the image of C,, has the following generator matrix :

[ aocl-1a-2.. . u-e,+l 12 where uvk is regarded as an nm-dimensional column vector over GF(q). This matrix is the generator matrix of C, itself. It can easily be shown that P,=-, contains no M- sequence code of length e, except for the images of y/I-array code. From this, the converse is apparent. Q.E.D.

Thus, we can construct y/I-arrays from M-sequences of length q”” - 1 over GF(q) by applying the inverse mapping 0f e.

IV. THE AUTOCORRELATION FUNCTIONS OF ~/?-ARRAYS

The autocorrelation function of the y/l-array has many important properties from the viewpoint of practical applications. First, the autocorrelation function of two-dimensional arrays is defined. Let 4 be a mapping from the direct product of GF(q) into the set of all real numbers:

4: (a,@ - 4(a,b),

where 4 is assumed to satisfy the three conditions:

4(G) = W ,a)

d(v) > 0

(32)

(33)

(34)

The actual form of the function 4 will be determined by the physical meaning of the elements of GF(q). In most cases, however, the previously mentioned restrictions will be adequate.

The autocorrelation function p+(z, ,z,) of an array a(x, y) is defined by

px--l Py-1

P4(71372) = igo j& ~(ai,j,ai+rl,j+r2)lS, (35)

where (p,,p,) is the period of the yfl-array and the subscripts u,v of au.” are calculated modulo pX and p,,, respec-

781

tively. Here s is given by px-1 Pr-1

,y = igo j50 4(ai,j~ad

From (33), s is always positive. Let us define x(a) for a E Wq) by

x(a) = J& 44h4 a E GE’(q). (36)

Further let a=q “m-ix(l) - 4(0,0). (37)

Then, the autocorrelation functions of y/&arrays can be given by the following theorem.

Theorem 10: The autocorrelation function pg(z1,z2) of a y/3-array in a period (i.e., for 0 I r1 < p,., 0 I z2 < p,) is given as follows.

If

21 = -qlk + k-,el mod pX,

z2 = N,,k + tc,el mod p,,, 0 I k < (q” - 1)//A o<z<q-1

(38) then

PJ7bZ2) = (4 run- ‘x(6’) - d@,0>Y~,

where ICY and ~~ are the integers satisfying

Nylcl + qk2 = 1 mod (4”“’ - 1) (39)

e is given by e = (q”” - l)/(q - 1) and 6 is a primitive element of GF(q) that is defined by 6 = c?. For values of z1 and z2 other than those of (38)

Remark: From (12), gcd (N,,yA) = 1. Thus integers K~ and ICY satisfying (39) can easily be obtained using Euclid’s algorithm.

Proqf: From Corollary 2, it is easily shown that

s = (4 ““-‘x(1) - wma7” - 1)/d. (41)

Then, p#(r,,z,) of (35) is rewritten as px-1 PY-1

p&,,zz) = igo jgo 4<f0,di~“~f,iZ>Is,

where f(‘,j)(x, y) was defined previously in (29) and fk,l(i’j) is the coefficient of the term x”y’ of ,f’i3j)(x,y). From Corollary 1, (37), and (41) we have

N,-1 N,-1

(42)

Suppose y”fi” E GF(q); then from Lemma 2

Thus


From Corollary 2 and (36), it can easily be shown that pg(zl ,T~) becomes

P&J,) = (4 “m- ‘x(y”P’“) - ~(O,O)}/o.

Now, let us obtain the r1 and z2 that satisfy y”fir2 E GF(q). If

y”p” = 61 = g’, 012<q-1

then Nyzl + $T, = el mod (4”” - 1). (43)

Multiplying both sides by K~ and using (39) we have

71 - ~/ZK,Z, + ~~qA.t.2 = rc,el mod (4”” - 1).

From this equation

Tl = K,el mod ,?

because 1, 1 q”“’ - 1. Similarly

z2 = K,el mod NY.

Then let z1 = rc,eZ + ?,k, and z2 = K,el + NykZ. Thus from (43)

k, = -vk, mod (q” - I)/&

Consequently, setting k, = k, we can obtain (38) immediately. For z1 and z2 other than (38), y”/3’* 6 GF(q) and, in this case, it is obvious that we can choose nm elements of the generator matrix G of C,, including g0,0 (E GF(q)) and gT,,r2 (= y-“p -‘* $ GF(q)) that are linearly independent over GF(q). Consequently, from Lemma 6, in the set {(fo,o(i.j) (id ,L,,,,) I 0 I i < N,, 0 I j < NJ VW) appears qnmM2 - 1 times and every other combination of two elements of GF(q) appears qnmm2 times. Now from (34) and (42), (40) is easily derived. Q.E.D.

This theorem indicates the following properties of the autocorrelation function of the y/3-array.

i) p+(r1,z2) assumes at most q values. ii) In the region 0 < r1 < N, and 0 I z2 < NY, the

number of points where the value of p+(z,,s,) is not equal to -4(0,0)/a does not exceed q - 1.

iii) Let (zr”,rzo) be an arbitrary element of the set T2 that is defined as the set of all (z,,z,) satisfying (38). Then the set {(zr - zr”, z2 - rzo) 1 (~i,z~) E T2} is equal to T2, if (z1,z2) is calculated module (p,,p,). In this sense, the points that have values of pg(r,,z2) not equal to -$(0,0)/o are uniformly distributed in the y/I-array.

Example: Let q = 2, $(O,O) = &l,l) = 1, and $(l,O) = 4(0,1) = - 1, then x(1) = 2, e = 0 mod (2”” - I), 6 = 1, and IS = 2”” - 1. Consequently,

P4(71?72)

ZZ.Z

L

1; T1 = -qik mod pX, z2 = N,k, 0 5 k < (q” - l)/pn

-l/(2”” - 1); in all other cases. (44)

V. SOME OTHER PROPERTIESOF@ARRAYS

The y/&arrays have many other interesting properties. In this section some of them are stated without proof. It is

not difficult to prove these properties, if one recalls the method of constructing $-arrays. These properties are closely analogous to those of one-dimensional M-sequences c71. A Relation Between yb-Array Codes and Cyclic Product Codes

A direct product code is a kind of two-dimensionally arrayed code. The following theorem makes clear the relation between a y/?-array code C,, and a cyclic product code [8].

Theorem 11: Let the minimal polynomials of y and /I over GF(q) be denoted by h,(x) and hs’(y), respectively. Let cyclic codes C, and C,’ of lengths pX and p,, be defined by parity-check polynomials h,(x) and h,‘(y), respectively. Then the yfi-array code C,, is a subcode of the cyclic product code C, x C,‘. Further, if the relation C,, = C, x C,’ holds, pX must be a divisor of q - 1.

The Macroscopic Parameters and the Symmetric Images of a y/&Array

It has been shown that an arbitrary pX x pY subarray of a y/I-array is constructed of p,/N,, = (q” - l)/pL/z maximum-area matrices (cf. Lemma 3). The parameters px,pY, N,,N,, and the mutual positions of pJN,, maximum-area matrices (i.e., the magnitude of the cyclic shift along the x axis, -ql mod pX, of two adjacent matrices) are called the macroscopic parameters of the yfi-array (or of the yfi-array code). An example of the macroscopic parameters of a y/3-array is shown in Fig. 5.

If among the parameters n, m, p, A, y, and a that deter- mine the yg-array code, n, m, p, and A are given, then pX, pY, N,., and NY are determined uniquely [cf. (21) and (26)]. Moreover, if y is given in addition, the mutual positions of the p,/N, maximum-area matrices are determined uniquely (cf. Lemma 3). The primitive element CI has no effect on those macroscopic parameters. If n,m are equal and p and/or 2 are different in two y/?-array codes, their macroscopic parameters cannot be the same. The macroscopic parameters are not necessarily different, however, when only q is different. Hereafter the y/?-array code having the parameters n, m, p, 2, q, and a is denoted by C: (n,m,/&l,u).

The following theorem shows the relation between two y/I-array codes whose macroscopic parameters are the same.

Theorem 12: A necessary and sufficient condition that a y/&array code C’: (n,m,pJ,~,a”) coincides with the y/I- array code C: (n,m,@.,q,u) is that y = q’ mod (q” - l)/ ~1 and v is given by

v = qk[(l + U.-l(q’q-’ - I)] mod (qnm - 1)

where k is an arbitrary positive integer and A-’ and y-l are integers satisfying ;iX ’ = 1 mod NY and yq- ’ = 1 modp,, respectively.

A yp-array and its symmetric images with respect to the x- and/or y-axes may generally be regarded to be the same, since all of them can easily be obtained if any one of them

NOMURA et al. : TWO-DIMENSIONAL LINEAR ARRAYS 183

\I 3001000 11011111100111 1010010 0110010 1100000

Fig. 5. Example of the macroscopic parameters of a yg-array (q=2, n=6, m=l, p=l, I=3, 4=2, N,=p,=9, py = 21, NY = I, --+I modp, = 3).

is given. Concerning the symmetric images of a y/I-array, we have proved the following theorem.

Theorem 13: The two-dimensional code C, that is the symmetric image of a yj-array code C, : (n,m,p&,a) with respect to the x axis is a yp-array code and is represented by G: <wvAe, - w).

The two-dimensional code Cc that is the symmetric image of C, with respect to the y axis is a y/?-array code and is represented by Cc: (n, m, p, 1, e, - II, u-l), where em = q”” - 1 is the order of LX

Corollary 3: The two-dimensional code CD that is the symmetric image with respect to both x and y axes of the code C, is a y/I-array code and is represented by CD: <vw,bw-l).

The Characteristic Arrays and the Coset Decomposit ion of the y/I-Array

Let f(x,y) be a yg-array whose period is (p,,p,). Then let qX and qY be integers satisfying

gcd (.wd = 1, 1 5 4x < Px (45)

gcd (py,qy) = 1, 1 5 9y < Py. (46)

Let us define a set Q to be t and qY, namely

Q = {(q.x,qy) I

where Q, = {qX} and QY =

he set of all ordered pairs of qX

qx E Qm qy E QyS>

{qY} are the sets of all qX and qY satisfying (45) and (46) respectively. Obviously Q has $~(p,)&,) elements, where 4 is Euler’s function.

A transformation rr(q,,qJ of a y/I-array is defined by

~(4X~9,u-(X> Y) = .Hxqx, Y”“) (47)

and is called a decimation of a yp-array. From (45) and (46) it is easily shown that a decimation is equivalent to a permutation of the coefficients of the y/I-array.

The following theorem forms the foundation of this section.

Theorem 14: n(q,,q,)f(x,y) is a y/?-array whose parameters n, m, p, and /1 are the same as those off(x,y).

Next, the characteristic arrays of a y/3-array are studied. Lemma 1.5: The macroscopic parameters of the yp-array

f(x,y) and the y/?-array rc(q,,q,)f(x,y) are the same if and

only if 4x -l = vmodp,

qY -’ = v modp,,

where v is some integer relatively prime to q”“’ - 1, and 4x-l and qydl are the integers defined by qXmlqX = 1 modp, and qY -lq,, = 1 mod py, respectively. There are m”m - 1) decimations that do not alter the macroscopic parameters of the y/I-array.

Lemma 26: The y/&array n(q,,q,,)f(x,y) is a cyclic shift of the yfl-array f(x,y) if and only if

9x-l = qk mod pX

9Y -’ = qk modp,,

where k is some positive integer. There are nm such decimations.

The yp-array whose image with an arbitrary decimation of Lemma 16 is the original array itself is called a characteristic array.

Theorem 27: There are q - 1 characteristic arrays among all q”” - 1 cyclic shifts of a yfl-array.

Theorem 18: The array that is obtained by a decimation of a characteristic array of a yb-array is also a characteristic array of a y/I-array.

Hereafter, qX and qy are regarded as representing reduced residue classes modulo pX and py, respectively. Then Q is a direct product of multiplicative groups Q, = {qX} and Q,, = {qy}, and is a commutative group with multiplication (qX,qy) * (qX’,qy’) = (qXqX’,qyqy’). The transformation group {z(q,,q,)} is isomorphic to Q when multiplication is defined as successive transformations.

Let C, be a set of the elements of Q that corresponds to a decimation that yields a cyclic shift of the y/I-array (cf. Lemma 16). Obviously C, is a subgroup of Q. Similarly, Co’ is defined to be a set of the elements of Q that corresponds to a decimation that does not alter the macroscopic parameters of the y/I-array (cf. Lemma 15). C,’ is a subgroup of Q. Furthermore, C, is a subgroup of Co’.

Theorem 19: A yp-array z(q,,q,)f(x,y) is a cyclic shift of a yb’-array 4q,‘,q,‘)f(x,y) if and only if (qx,qy) and (qx’,qy’) belong to the same residue class of a subgroup C,, of Q, where f(x,y) is a y/?-array. Further, the macroscopic parameters of n(q,,q,Mx,y) and 4q,‘,qy’Vhd are the same if and only if (q,.,q,,) and (qX’,qY’) belong to the same residue class of a subgroup C,’ of Q.

Theorem 20: Let Q’ = {(i,j) 1 0 < i < pX, 0 5 j < p,}. The element (i,j) is regarded as representing the (i,j)th position of a y/J’-array. Let Q’(I Q) be decomposed into residue classes (including improper cosets) of the subgroup C, of Q. Then the positions of a characteristic array of a yfl-array have the same coefficient (element of GF(q)) if they belong to the same residue class.

Theorem 22 : If a yp-arrayf’(x, y) has the same parameters n, m, 11, and 1, as a yfl-arrayf(x, y), we can constructf’(x, y) from f(x,y) by applying a decimation and executing a cyclic shift.

Theorem 22: When parameters n, m, p, and a satisfying conditions (IO), (1 l), and (12) are given, there are

184 IEEETRANSACTIONSONINFORMATIONTHEORY,NOVEMBER 1972

TABLE I PARAMETERS OF THE BP-ARRAYS OVER GF(2) WITH AREAS N,N, NOT

GREATERTHAN 1023 AND NOTLESSTHAN 3

TABLE I (Cont’d.) PARAMETERS OF THE yj3-ARRAYS OVER GF(2) WITH AREAS N,N, NOT

GREATERTHAN 1023 AND NOTLESSTHAN 3

+r

: 1

7x 3 1 1

In 4 2

: 1 1 1 1 1

*I 5 1 1

I* 6 3 2 2 1 1 1 1 1 1 1 1 1 1 1

*I 7 1 1

m a 4 4 2 2 2 2 2

: 1 1 1 1

:

: 1 1 1 1 1 1 1 1 1

m 9 3 3

: 1 1 1 1

P 1 1 1

A 1

:

A 1 1 1 1 1 1 1 3 3

r* 1 1 1

fi 1 1

: 1 1

: 1 1 3 3 7 7 7

.- 1 1 1

< 1 1 1 1 1 1 3 3 1 1 1 1 1 1 1 1 3 3 3 3 5

: 5

15 15

Y 1 1 1

: 1 7 I

h 1 1 3

IeNx P) 1 3 3 3 3 1

NY 3 1 1

R. 1 1 7

h'N. ry 1 7 7 7 1 1

NY 7

:

h 1 1 3 1 3

1: 1 5

P..NI 0 1 15 3 15 3 5

15 15 15 5 15 3 15 1

5 15 5 3

4 15

5 5 1 1 1 1 3 3

x

: 31

?.s'Nl. ry

3: i: 31 1

N? 31

1 1

h h-h. ca 1 1 63 1 3 63 1 7 63 7 7 9 1 63 63 3 63 21 7 63 9 9 63 7

21 63 3 63 63 1

1 21 63 7 21 9 1 9 63 3 9 21 9 9 7

NY pr h 63 1 21 3

9' : 1 63 1 21 1 9 1 I 1 3 1 1 3 21 3 3 7 9 7 3 7 1

A Peu* Pq 3 1 1 127 127 1 127 127 1

127 127 1 1

?. 1 1 3 1 3 5

15 1 5 1 3 5

15 17 51 a5

255 1 5

17 a5

1 3

17 51

1:

h4k t? 1 255 3 255 3 a5

15 255 15 a5 15 51 15 11

5 255 5 51

255 255 255 a5 255 51 255 17 255 15 255 5 255 3

255 a5 25: a5 a5 :: 5": 255 3

2 2 51 17 25," 17 15

"? 255

a5 a5 17 17 17 17 51 51

: 1 1

: 1 1 3 3

i 5 6 5 5

15 15

h 1 1 7 1 7

73 511

1 73

h.dx 1 5% 7 511 I 73

511 511 511 73 511 7

511 73 51: 73 7

5:: 73 73

: 1 1 7 7

h IN) 1 3 1

h hj 1 7 1

h/Ivy 1 3 1

15 5 3 1

:

*I* 1

31 1

'Y/N]

12: 1

'F 3 1

15 5 3 1 5 1

255 a5 51

17 15

5 3 1

a5 17

5 1

51 17

3 1

17 1

h/N 1)

: 611

73

: 73

1

nm = 10 n fi i 1: 1 2 5 1 2 5 1 5 2 1 5 2 1

10 1 1 10 1 1 10 1 1 10 1 1 10 1 1 10 1 1

WM. 1 3 3

31 31

1023 1023 1023 1023 1023 1023 1023 1023

341 341 341 341

h Y vi 1023 1023 1 1023 341 3

341 341 1 1023

33 i: 31 1 1023 1 1023

341 1 341 93 93

33

:

31 1 :: 11 1 11

1 3 : 1023

93 33

3 1023

341 33

1 1 3 341 3 31 3 11 3 1

11 93 11 31 11 3 11 1 31 33 31 11 31 31 :

Ml-2 s A

N*

M*

M*

M*

M*

M

M*

M

M'

M*

M*

M

M*

M

M*

M*

M*

M

M

M

l4*

M

n

M

M*

M

M*

M

1 1 3 1

31

lx*

n

M

II*

M

H

M

H

; rim = 3

I 1 3 3

nm = 4 *

1 3

11 31 33 93

341 1023

1 11 31

341 1

10 1 1

: 2

E 1 1

10 1 10 1 10 1 10 1

1 3 3 3 3

11 11

4 4 4 4 4 4

nm = 5 n

93 93 10 1 3

31 93

1

1: 33

10 1 10 1

:: 1 1

10 1 10 1

11 11 31 31 31

93 93 33 33 33 33

11 1023

341 93 31 1

5 31 93 93

10 1 10 1

1 11 1023 93 11 11 11 93 93 1 5

nm = 6 n 1 2

z 6 6 6 6 6 6

i 6 6 6

nm = I II

An M in the last column indicates that the r8-array is a r8-M-array, and M* indicates that a period of the y8-array is an M-sequence.

and the number of y/Carrays is

(q”” _ 1) d4qnm - 1) nm

Remark: It can be shown that 1 7

nm = a n 1 2 2 4 4

z 4 4 a a

VI. CONCLUSIONS

We have given a method of construction and many interesting characteristics of two-dimensional linear recurring arrays having maximum-area matrices. Most of the characteristics of the y/?-arrays we have described can be regarded as two-dimensional analogs of those of the usual M-sequences. The characteristic peculiar to y/Sarrays that is not present in M-sequences is the difference between the two concepts of period and area of the maximum-area matrices.

A problem left unsolved is whether there exist arrays having maximum-area matrices other than y/Larrays. No such example has been found so far, though it has not been proved that y/I-arrays exhaust all of the linear recurring arrays having maximum-area matrices.

ACKNOWLEDGMENT

The authors wish to thank Prof. Y. Taki, Prof. Y. Yasuda, and Dr. K. Ito of the University of Tokyo for their very stimulating discussions and encouragement during the course of this study.

APPENDIX

The parameters of the yfi-arrays over GF(2) with area N,N, not greater than 32 767 and not less than 3 have been calculated

nm q 9

‘: 3

9 9 9

c$[(q” - l)/pn] y/Sarray codes whose macroscopic structures are different. The number of y/?-array codes is

4(qflrn - 1) nm 4 (pf)

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-18, NO. 6, NOVEMBER 1972 785

by digital computer. The y/3-arrays with area not greater than 1023 are tabulated in Table I.

REFERENCES [l] I. S. Reed and R. M. Stewart, “Note on the existence of perfect

maps,” IEEE Trans. Inform. Theory, vol. IT-8, pp. 10-12, Jan. 1962. [2] R. Spann, “A two-dimensional correlation property of pseudo-

random maximal length sequences,” Proc. IEEE (Corresp.), vol. 53, p. 2137, Dec. 1965.

[3] B. Gordon, “On the existence of perfect maps,” IEEE Trans. Inform. Theory (Corresp.), vol. IT-12, pp. 486-487, Oct. 1966.

[4] T. Nomura and A. Fukuda, “Linear recurring planes and two-

[51

I61

t71

I81

[91

dimensional cyclic codes,” Trans. Inst. Electron. Commun. Eng. Jap., vol. 54-A, pp, 147-154, Mar. 1971. E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw- Hill, 1968. T. Kasami, S. Lin, and W. W. Peterson, “New general ization of the Reed-Muller codes, Part I: Primitive codes,” IEEE Trans. Inform. Theory, vol. IT-14, pp. 189-199, Mar. 1968. S. W. Golomb, Shift Register Sequences. San Francisco: Holden- Day, 1967. H. 0. Burton and E. J. Weldon, “Cyclic product codes,” IEEE Trans. Inform. Theory, vol. IT-l 1, pp. 433-439, July 1965. D. Calabro and J. K. Wolf, “On the synthesis of two-dimensional arrays with desirable correlation properties,” Inform. Contr., vol. 11, pp. 537-560, Nov./Dee. 1967.

On the Number of Information Symbols in Polynomial Codes

SHU LIN, MEMBER, IEEE

Abstract-Polynomial codes and their dual codes as introduced by Kasami, Lin, and Peterson have considerable algebraic and geometr ic structure. It has been shown that these codes contain many well-known classes of cyclic codes as subclasses, such as BCH codes, projective geometry codes (PG codes), Eucl idean geometry codes (EG codes), and general ized Reed-Muller codes (GRM codes). In this paper, combinatorial expressions for the number of information symbols and parity- check symbols in polynomial codes are derived. The results are appl ied to two important subclasses of codes, the PG codes and EG codes.

I. INTR~DUCTI~N

P

OLYNOMIAL codes and their dual codes as introduced by Kasami et al. [l] h ave considerable inherent math-

ematical structure [l]-[5]. It has been shown [l], [2] that these codes contain several important classes of cyclic codes as subclasses, such as BCH codes [6], [7], finite projective geometry codes (PG codes) [8]-[12], Euclidean geometry codes (EG codes) [8], [ 1 l]-[13], and generalized Reed- Muller codes (GRM codes) [14]. In this paper, the problem of enumerating the number of information symbols in polynomial codes and their dual codes is studied. First, combinatorial expressions for the number of information symbols and the number of parity-check symbols in polynomial codes are derived. Next, we apply these results to projective geometry codes and Euclidean geometry codes.

In order to present our results, it is necessary to give a brief review of polynomial codes and their duals. For details, the reader is referred to [I]-[4].

Let m and s be two positive integers. Let GF(q”“) be the extension field of the field GF(q”), where q is a power of a

prime p. Let c1 be a primitive element of GF(q”“). Any nonzero element c(j in GF(qms) can be expressed as

cd = aIj + azjal + asjcc2 + ... + amjurn-l (1)

for 0 I j < q”” - 1, where aij is in GF(q”). There is a one-to-one correspondence between clj and the m-tuple Aj = (aIj,azj;. . ,u,~). The m-tuple Aj is called the coordinate vector of c&. Let b be a factor of q” - 1. We define two positive integers, z and n, as follows:

z = (q’ - 1)/b (2)

n = (4”” - 1)/b. (3)

For the purposes of clarity, the definitions of b, z, and n will be recalled frequently throughout this paper.

Let X = (X1,X,;. .,X,), where Xi is a variable over GF(q”). Let p be a positive integer less than mz. Define Q,(p,b) as the set of polynomials in m variables,

j-(X) = c c * * * c c,,,,. . .ymx1v’x2v* . . * X,‘- (4) VI Y2 Vm

for which

1) G,,, vm 6 GFW)

2) 0 2 vi < qs, l<ilm

ill Vi = lb, 0 I 1 I ~

4) f(alj,azj,. . . ,amj) E GF(q), Oljcn,

where (a,j,a, j,. . . , a .) is the coordinate vector of crj. mJ For each polynomial f(X) in Q,(p,b), a vector v(f) is defined as follows:

Manuscript received November 1, 1971; revised April 10, 1972. This work was supported by NSF Grant GK-25128.

The author was with the University of Hawaii, Honolulu, Hawaii. He is currently on, leave at the Department of Electrical Engineering, University of Utah, Salt Lake City, Utah 84112. ._ _ - v(f) = (wJl,U2,‘~ .,%I), (5)

a theory of two-dimensional linear recurring arrays

Documents