IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 5, SEPTEMBER 1997 1613

Characterization of QuaternaryExtremal Codes of Lengths and

W. Cary Huffman

Abstract—We prove that, up to equivalence, there is a unique extremalquaternary Hermitian self-dual code of length18, and that there are twoinequivalent extremal quaternary Hermitian self-dual codes of length20.

Index Terms—Extremal codes, Hermitian self-dual codes, quaternarycodes.

I. INTRODUCTION

In this correspondence, we will classify the extremal Hermitianself-dual codes over GF(4) of lengths18 and 20. We begin withnotation and background material.

An [n; k] (linear quaternary) code C over F4 = GF(4) is ak-dimensional subspace ofFn

4 where F4 = f0; 1; !; !g with! = !2 = ! + 1. The weightwt (ccc) of a codewordccc 2 C is thenumber of nonzero components ofccc. The minimum nonzero weightd of all codewords inC is called theminimum weightof C; an [n; k]code with minimum weightd is denoted an[n; k; d] code. We willendowFn

4 with the Hermitian inner producth�; �i given by

huuu; vvvi =

n

i=1

uiv2

i

whereuuu = u1u2 � � � un and vvv = v1v2 � � � vn are in Fn

4 . Thedual of C is the [n; n � k] codeC? defined by

C? = fvvv 2 F

n

4 jhuuu; vvvi = 0 for all uuu 2 Cg:

The codeC is (Hermitian) self-orthogonalif C � C? and (Hermitian)self-dual if C = C?. If C is self-dual, thenk = 1

2n. By [7], the

minimum weight of an[n; 1

2n; d] quaternary code satisfies the bound

d � 2bn=6c+2. The self-dual code isextremal1 if this bound is met.So extremal self-dual codes of lengths18 and 20 have minimumweight 8. The number of codewords inC of weight i is denotedAi(C), andfAi(C)j0 � i � ng is the weight distributionof C. IfC is a self-orthogonal quaternary code, thenC has only even-weightcodewords and soAi(C) = 0 if i is odd. Thesupport of a vectoristhe set of coordinates on which the vector is nonzero; thesupport ofa codeis the union of the supports of the codewords in the code.

The extremal quaternary codes have been classified up to length16 in [2] and [7] by completely enumerating all self-dual quaternarycodes of each length. Such complete enumeration seems infeasiblefor lengths greater than16. In [6], the nonexistence of an extremalself-dual quaternary code of length24 was verified, without resorting

Manuscript received May 17, 1996; revised January 27, 1997. The materialin this correspondence was presented in part at the 34th Annual AllertonConference on Communication, Control, and Computing, Monticello, IL,October 2–4, 1995.

The author is with the Department of Mathematical Sciences, LoyolaUniversity, Chicago, IL 60626 USA.

Publisher Item Identifier S 0018-9448(97)05218-8.1These extremal codes are also called extremal Type IV codes.

to complete enumeration. In [4] and [5] it was shown that up toequivalence there is exactly one extremal self-dual quaternary codeof length 18 which possesses a nontrivial automorphism of oddorder. This code is denotedS18 in [7] and is thoroughly describedin [1]. In the next section, we establish thatS18 is the unique[18; 9; 8] self-dual quaternary code, up to equivalence. It was alsoshown in [4] and [5] that there are exactly two inequivalent extremalself-dual quaternary codes of length20 which possess a nontrivialautomorphism of odd order greater than3 and two inequivalentextremal self-dual quaternary codes of length20 which possess anontrivial amorphism of order3; the relationship of the first twocodes to the last two was left undetermined. In Section III of thiscorrespondence we show that there are exactly two inequivalent[20; 10; 8] self-dual quaternary codes. In the last section we presenta table summarizing information known about extremal Hermitianself-dual quaternary codes of length up to30

LetMn be the group of alln�n monomial matrices overF4. LetGal (F4) = f1; �g be the Galois group ofF4 where� : x 7! x2 is theconjugation map. LetWn be the semidirect product ofMn extendedby Gal (F4); an elementT 2 Wn can be written asT = PD� whereP is a permutation (matrix),D is a diagonal matrix with nonzerodiagonal entries, and� 2 f1; �g. CodesC1 and C2 of length n

are equivalentprovided there exists an elementT 2 Wn such thatC2 = C1T . The automorphism groupof the lengthn codeC is thegroupAut (C) =fT 2 Wn j CT = Cg.

II. THE CASE n = 18

Let C be an[18; 9; 8] self-dual code. Let

C0 = fc1c2 � � � c18 2 C j c17 = c18g:

As C = C? has no wight2 codewords,C0 6= C and henceC0 is8-dimensional. Let

D = fc1c2 � � � c16 j c1c2 � � � c18 2 C0g

be C0 punctured on coordinates17 and 18. ThenD is a [16; 8; d]

self-dual code withd � 6. By the classification in [2],d = 6 andDis one of four codes with generator matrices given in [2, figs. 5 and 6]denoted #52–#55. The generator matrices for #53–#55 have all rowsof weight6 and such a generator matrix can be found for #52 as well.In the following, letD be a generator matrix forD with all rows ofweight 6. Therefore, there is a generator matrix ofC0 of the form

gen (C0) = D

a1 a1

a2 a2...

...a8 a8

(1)

wherea1; � � � ; a8 2 f1; !; !g. By rescaling columns17 and 18 ofC we may assumea1 = 1.

The monomial partMAut (D) of Aut (D), consisting of theelements fromAut (D) of the formPD� with � = 1, is determinedin [2]. By allowing conjugation, we can findAut (D), which is twicethe size ofMAut (D). For each of the four choices forD all 37

0018–9448/97$10.00 1997 IEEE

1614 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 5, SEPTEMBER 1997

Fig. 1. Generator matrices for the two possibilities forC0.

possibilities fora2; � � � ; a8 were examined by computer; the numberof possibilities is decreased dramatically by applying elements ofAut (D) to the first 16 coordinates of (1) and reducing to the sameform. When the codeD is #52–#55, we reduced (1) to one of 2,24, 9, and 63 possible forms, respectively. Of the codes generatedby these 98 matrices, only two actually had minimum weight8, oneusing code #53 and the other #54. Generator matrices for these twocodes, written in reduced echelon form and denotedC0

0 andC00

0 aregiven in Fig. 1.

One easily determines thatC0?

0 is generated by the eight rows ofgen (C00)

vvv0

9 =01!0!0!1!000000010

and

vvv0

10 =000000000000000011:

If C0 = C00, C is generated by the eight rows ofgen (C00) and eithervvv09 + !vvv010 or vvv09 + !vvv010, as C0 � C � C?0 and C has minimumweight 8. These two choices lead to equivalent codes by switchingcoordinates17 and18. The code thus generated in fact has minimumweight8. With either choice, the codeC has automorphismPD where

P =(1; 2; 4; 3; 5)(7; 8; 11; 9; 10)(12; 13; 16; 14; 15)

and

D =diag (1; 1; 1; 1; 1; 1; 1; !; !; !; !; 1; !; !; !; !; 1; 1):

By [4, Theorem 5],C is equivalent toS18.Similarly, C00?0 is generated by the eight rows ofgen (C000 )

vvv00

9 =110010111000000010

and

vvv00

10 =000000000000000011:

If C0 = C000 , C is again generated by the eight rows ofgen (C000 )

and eithervvv009 + !vvv0010 or vvv009 + !vvv0010, choices leading to equivalentcodes by switching coordinates17 and18. This [18; 9; 8] code hasautomorphism

P = (2; 6; 5; 3; 4)(7; 14; 15; 16; 9)(8; 12; 11; 10; 13):

By [4, Theorem 5],C is again equivalent toS18. These argumentsgive the following theorem.

Theorem 2.1: If C is an [18; 9; 8] self-dual quaternary code, thenC is equivalent toS18.

III. T HE CASE n = 20

The technique of the previous section cannot be used withn = 20

because the[18; 9; 6] self-dual quaternary codes have not beenclassified. Instead, we approach the problem as in [6].2

Let C be an [n; 1

2n] self-dual code. Fixn1 and n2 so that

n1 + n2 = n. Let B, respectively,D, be the largest subcode ofC whose support is contained entirely in the leftn1, respectively,right n2, coordinates. SupposeB andD have dimensionsk1 andk2,respectively. Letk3 = 1

2n � k1 � k2. Then there exists a generator

matrix for C in the form

gen (C) =

B O

O D

E F

(2)

whereB is ak1�n1 matrix with gen (B) = [B O], D is ak2�n2

matrix with gen (D) = [O D], O is the appropriate size zero matrix,and [E F ] is a k3 � n matrix. Let B� be the code of lengthn1generated byB, BE the code of lengthn1 generated by the rows ofB andE, D� the code of lengthn2 generated byD, andDF thecode of lengthn2 generated by the rows ofD andF . The followingresult, whose proof is included for completeness, is found in [8].

Lemma 3.1: With the notation of the previous paragraph

i) k3 = rank (E) = rank (F ),ii) k2 = 1

2n + k1 � n1, and

iii) B?E = B� andD?

F = D�.

Proof: If i) fails, there is a nonzero codewordccc 2 C that is alinear combination of the rows of[E F ] which is zero on either thesupport ofB or the support ofD. By maximality ofB andD, ccc mustbe in eitherD or B, respectively, a contradiction as the rows of thegenerator matrix in (2) are independent. So i) holds.

By self-duality ofC, the rows ofB andE are orthogonal to therows of B. Thus

k1 + (k1 + k3) � n1: (3)

Similarly

k2 + (k2 + k3) � n2: (4)

Adding, we have2(k1 + k2 + k3) � n1 + n2 = n; but by definitionof k3, this inequality must be an equality and hence equality holdsin (3) and (4), proving iii). Part ii) follows from the equality in (3)and the definition ofk3.

Now assume thatC is a [20; 10; 8] self-dual code. LetB be thecode generated by a single wight8 vector, whose support we canplace in the first eight coordinates. Son1 = 8, k1 = 1, andn2 = 12

from which it follows thatk2 = 3 by Lemma 3.1 ii). Therefore,D�

is a [12; 3; d�] self-orthogonal code withd� � 8. We need to findthe possibilities forD�.

2Even though the technique is similar here to that used in [6] in anattempt to find a (nonexistent)[24; 12; 10] quaternary self-dual code, thereare increased complications, and hence additional techniques required, becausethe codes actually exist. To illustrate this complication, in Lemma 3.2 we showthat up to equivalence there are five possible 3-dimensional subcodes whosesupports are disjoint from the support of a given minimum-weight vector; in[6], the corresponding 4-dimensional subcode was unique. At each stage wewant to reduce the number of codes in the end which must be broken intoequivalence classes. So we must reduce, whenever feasible, the number ofpossible generator matrices.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 5, SEPTEMBER 1997 1615

Lemma 3.2: Up to equivalenceD� is a [12; 3; 8] code with oneof the following five generator matrices:

D1 =1 1 1 1 1 1 1 1 0 0 0 00 1 0 0 1 1 ! ! 0 1 1 10 0 1 ! 1 ! ! ! 1 1 ! !

D2 =1 1 1 1 1 1 1 1 0 0 0 00 1 0 0 0 1 ! ! 1 1 1 10 0 1 0 ! 1 1 ! 0 1 ! !

D3 =1 1 1 1 1 1 1 1 0 0 0 00 1 0 0 0 1 1 1 1 1 1 10 0 1 0 1 1 ! ! 0 1 ! !

D4 =1 1 1 1 1 1 1 1 0 0 0 00 1 0 0 0 1 1 1 1 1 1 10 0 1 0 ! 0 1 ! 1 ! ! !

and

D5 =1 1 1 1 1 1 1 1 0 0 0 00 1 0 0 0 1 1 1 1 1 1 10 0 1 0 1 0 1 1 ! ! ! !

:

LetD�

j be the code generated byDj . The weight distributionAi(D�

j )is given by

i 8 10 12

Ai(D�

1) 27 36 0Ai(D

�

2) 30 20 3Ai(D

�

3) 33 24 6Ai(D

�

4) 36 18 9Ai(D

�

5) 45 0 18

whereA0(D�

j ) = 1 and all otherAi(D�

j ) = 0.Proof: Let D be a generator matrix forD�. It is easy to show

that there is no[11; 2; 10] self-orthogonal code. This implies thatthe first two rows ofD can be chosen to be of weight8 with row1 equal to111111110000 and, up to equivalence, row 2 equal toone of0101!!!!0011 or the second row ofD1, D2, or D3. Thereare “obvious elements” ofW12 which fix the first two rows ofD.By computer, the third row ofD was determined, up to equivalenceunder these obvious elements. There were 112 possible third rowsleading to codes of minimum weight8; these 112 codes had fivedifferent weight distributions. By choosing a different weight8 vectorin each of these 112 codes as a new first row and row reducing, bycomputer it was shown that codes with the same weight distributionwere equivalent. The resulting codes and their weight distributionsare as stated.

We will say that two vectors arecomplementaryif their supportsare disjoint. A wight8 vector ism-complementaryif its support isdisjoint from the supports of exactlym wight 8 vectors. By Lemma3.2, a wight8 codeword ofC is m-complementary form = 27,30, 33, 36, or 45; we will ultimately show thatC contains a45-complementary wight8 vector.

For the five choicesD = Di in Lemma 3.2, we can find the matrixF = Fi of (2) by using Lemma 3.1 iii). These choices are given inFig. 2. We only need to construct the possible matricesE = Ei foreach i. For 2 � i � 5, the first row ofFi has weight2; usingorthogonality, the minimum weight, and possibly adding a scalarmultiple of our original wight8 vector inB, it is easy to see that row1 of Ei can be assumed to be11!!!!00. Similarly, with additionalinformation aboutAut (D�

1), row 1 of E1 can be assumed to be111!!000.

At this point, we have five possibilities for the first five rows ofgen (C). We can find elements ofW20 which fix the form of thesefirst five vectors and the second row ofFi. Using these to reduce thenumber of possibilities for row 2 ofEi, we obtain 10, 3, 4, 4, and

Fig. 2. Possibilities forF in (2) paired withD = Di.

4 possibilities for this row wheni = 1, 2, 3, 4, and5, respectively.We can reduce these 25 possibilities to 11 cases. To do this, we needthe following lemma. Its proof can be done directly by appealingto [3, Table II]; we choose to present a proof in the context of thecurrent problem.

Lemma 3.3: A self-orthogonal[12; 4; 8] quaternary code does notexist.

Proof: If such a code exists, by row reduction, there also existsa self-orthogonal[11; 3; 8] code with a wight8 vector. Decomposegen (C) as in (2) using this vector as the first row. The rest ofgen (C)can be completed according to Lemma 3.2 and Fig. 2. If the[11; 3; 8]subcode exists, we must be able to choose three more coordinatesfrom the right twelve coordinates to form, together with the left eightcoordinates, the support of the[11; 3; 8] code. By examining thefive possibilities from Lemma 3.2 and Fig. 2, this choice cannot bemade.

Corollary 3.4: If B is a 3-dimensional subcode ofC whose supportsize is12, there exists a wight8 codeword ofC whose support is thecomplement of the support ofB.

Proof: By Lemma 3.3,B is maximal of support size12. In thenotation of Lemma 3.1,D is 1-dimensional of support size8 and socontains the desired vector.

We define the following process based on this corollary. If thefirst row ccc1 of (2) is k-complementary, and if we can find a vectorccc0 2 C which ism-complementary withk < m, then we can replaceccc1 with ccc0 and reconstruct the generator matrix in the same form(2). The new codeD is uniquely determined by Lemma 3.2 andthe number of wight8 vectors complementary toccc0. We call thisprocess aswap; we will only swap when we increase the number ofcomplementary wight8 vectors. Potential vectors to use in a swapcome from Corollary 3.4. If there is a 3-dimensional subcodeD0

with support of size12, by the corollary, there is a wight8 vectorccc0

disjoint from this subcode; if the number of wight8 vectors inD0 isgreater than the number of wight8 vectors inD, then swapccc0 for ccc.

We have 25 possible cases for the first six row vectors of (2).The codeD0 generated by rows 1, 5, and 6 has support size12 byexamination ofDi in Lemma 3.2 andFi in Fig. 2. In 14 cases,D0

has more wight8 vectors thanD, and hence we can swap to one ofthe remaining 11 cases.

Possible vectors for row 3 ofEi (row 7 of (2)) were computedlexicographically for each of the 11 cases obtained so far. Reductionswere made by equivalence under certain maps ofW20 sometimes

1616 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 5, SEPTEMBER 1997

Fig. 3. First seven rows of (2) when row 1 is a45-complementary vector.

Fig. 4. Last three rows of generator matrices for[20; 10; 8] codes.

TABLE IINEQUIVALENT EXTREMAL HERMITIAN SELF-DUAL QUATERNARY CODES

combined with the addition of previously chosen vectors. Swappingalso reduced the number of cases. For example, when consideringE4, by looking atD4 and F4, we see that the codeD0 generatedby either rows 1, 5, and 7 or rows 1, 6, and 7 has support size12,and can potentially lead to a swap. Finally, the minimum weight ofthe code generated by the first seven vectors was checked to alsoreduce the number of cases. When these reductions were completedthere were 139 possibilities for the first seven rows of (2). It turnedout that there was only one case among these 139 whose first rowwas a45-complementary vector. The first seven rows of this caseare given in Fig. 3.

Applying similar techniques to the process of choosing rows 8 and9 of (2) yielded a total of 931 cases. Possibilities for row 10 weregenerated and reduced as before. Those codes of minimum weight8 which survived and whose first row of (2) wasm-complementaryfor m < 45 were examined for the possibility of swapping. Thistime, as the full code can be examined, we generated the3 � 570

wight 8 vectors and found in all cases either3 � 30 or 3 � 10 45-complementary wight8 vectors. Thus every[20; 10; 8] self-dual codehas a45-complementary wight8 vector. Furthermore, the first sevenrows can now be chosen to be those of Fig. 3. In the end, ten codes

were left. Of these ten, eight had3 � 540 33-complementary vectorsand 3 � 30 45-complementary vectors while the remaining two had3 �240 33-complementary vectors,3 �320 36-complementary vectors,and 3 � 10 45-complementary vectors. With the aid of a computerprogram (with results verified by hand), the first eight codes wereshown to be equivalent and the remaining two were equivalent. Thisproves that there is a unique code with3 � 30 45-complementaryvectors (generated, for example, by the seven rows of Fig. 3 andthe three rows ofG1 in Fig. 4) and a unique code with3 � 10 45-complementary vectors (generated using Fig. 3 andG2 of Fig. 4);these are, of course, inequivalent. By examining [4] and [5], eachcode has a nontrivial automorphism of order3 and an automorphismof order5. This verifies the following theorem.

Theorem 3.5: If C is a [20; 10; 8] self-dual code, thenC isequivalent to one of two inequivalent codes. Each of these codeshas a nontrivial automorphism of order3 and an automorphism oforder 5.

IV. CONCLUSION

Table I summarizes information currently known about extremalHermitian self-dual quaternary codes of lengths up to30. The columnlabeled “number” gives the number of inequivalent[n; n=2; d]

extremal Hermitian self-dual quaternary codes; the column labeled“status” tells whether or not the classification is complete and givesreferences for this information.

ACKNOWLEDGMENT

The author wishes to thank one of the referees for pointing out [3]and improving the proof to Lemma 3.3.

REFERENCES

[1] Y. Cheng and N. J. A. Sloane, “The automorphism group of an[18; 9; 8] quaternary code,”Discr. Math., vol. 83, pp. 205–212, 1990.

[2] J. H. Conway, V. S. Pless, and N. J. A. Sloane, “Self-dual codes overGF(3) and GF(4) of length not exceeding16,” IEEE Trans. Inform.Theory, vol. IT-25, pp. 312–322, 1979.

[3] P. P. Greenough and R. Hill, “Optimal linear codes over GF(4),” Discr.Math., vol. 125, pp. 187–199, 1994.

[4] W. C. Huffman, “On extremal self-dual quaternary codes of lengths18to 28, I,” IEEE Trans. Inform. Theory, vol. 36, pp. 651–660, 1990.

[5] , “On extremal self-dual quaternary codes of lengths18 to 28, II,”IEEE Trans. Inform. Theory, vol. 37, pp. 1206–1216, 1991.

[6] C. W. H. Lam and V. Pless, “There is no(24; 12; 10) self-dualquaternary code,”IEEE Trans. Inform. Theory, vol. 36, pp. 1153–1156,1990.

[7] F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward,“Self-dual codes over GF(4),” J. Combin. Theory A, vol. 25, pp.288–318, 1978.

[8] V. Pless, N. J. A. Sloane, and H. N. Ward, “Ternary codes of minimumweight 6 and the classification of the self-dual codes of length20,”IEEE Trans. Inform. Theory, vol. IT-26, pp. 305–316, 1980.