Information and Coding Theory Autumn 2017

Homework 4Due: December 6, 2017

Note: You may discuss these problems in groups. However, you must write up your own solutionsand mention the names of the people in your group. Also, please do mention any books, papers orother sources you refer to. It is recommended that you typeset your solutions in LATEX.

1. More on linear codes. Recall that a linear code C : Fkp → Fn

p was specified by agenerator matrix G such that ∀x ∈ Fk

p, C(x) = Gx. The parity-check matrix was amatrix H such that the columns of HT form a basis for the null-space of GT. Provethe following facts about linear codes.

(a) Prove that for a linear code C, the distance ∆(C) can be written as

∆(C) = minz∈C\{0n}

wt(z) ,

where 0n denotes the all-zero vector in Fnp and wt(z) denotes the number of

non-zero entries in z.

(b) Let n = 2r − 1 for some integer r. Recall that the general Hamming code (overthe field F2) is defined by the parity-check matrix H ∈ Fr×n

2 where the ith col-umn of H is given by the number i written in binary using r bits (take the topentry to be the most significant bit and the bottom entry to be the least signifi-cant bit). Find the message length, block length and the distance for this code.

(c) For a linear code C with generator matrix G and parity-check matrix H, it’s dualcode C⊥ is defined as a code with generator matrix HT. Prove that GT is a parity-check matrix for C⊥. Find the message length, block length and distance for thedual code of the Hamming code defined above.

2. Codes and pseudorandomness. In this problem, we will use codes to construct pseu-dorandom objects known as t-wise independent distributions. Let C : Fk

2 → Fn2 be a

linear code with distance ∆(C) = d, and let H ∈ F(n−k)×n2 be the parity-check matrix

of this code.

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(a) First consider z uniformly distributed in Fn−k2 . Using the fact that z is a random

binary string of length n− k, prove that for any a ∈ Fn−k2 \ {0n−k}

Ez∈Fn−k

2

[(−1)a·z] = 0 where a · z = aTz =n−k

∑i=1

aizi mod 2 .

(b) Prove that the code can be used to extend this property of the uniform dis-tribution over length n − k strings, to a distribution over n bits i.e., we can“stretch” the pseudorandomness. Consider the distribution obtained by choos-ing z ∈ Fn−k

2 at random and taking x = HTz. Note that x ∈ Fn2 . Prove that for

any b ∈ Fn2 \ {0n} with wt(b) < d, we have

Ez∈Fn−k

2

[(−1)b·x

]= E

z∈Fn−k2

[(−1)b·(Hz)

]= 0 .

Such distributions are called (d− 1)-wise independent distributions on n bits,since they “look like” the uniform distributions as long as one looks at at most(d− 1) bits at a time.

(c) Show that the Hamming code can be used to produce a 2-wise independentdistribution on n = 2r − 1 bits, starting with the uniform distribution on just rbits.

3. Scrambled Reed-Solomon Codes [due to Venkat Guruswami]. Let {a1, . . . , an} bedistinct elements of Fp used to define a Reed-Solomon code C : Fk

p → Fnp. Assume

that k < n/6. Recall that a message (m0, . . . , mk−1) is encoded by thinking of it as apolynomial P(X) = ∑k−1

j=0 mj · X j and sending (P(a1), . . . , P(an)). For the followingparts, assume the fact (used in class) that for a bivariate polynomial Q(X, Y), we canfind all its factors of the form Y− f (X).

(a) Suppose we sent two codewords according to the polynomials P and P′ (of de-gree k− 1) but they got mixed up. Thus, we now have two lists (b1, . . . , bn) and(c1, . . . , cn) and we know for each i ∈ [n]

either P(ai) = bi and P′(ai) = ci or P(ai) = ci and P′(ai) = bi

Note that each coordinate could be independently scrambles i.e., it may hap-pen that for some i, P(ai) = bi and P′(ai) = ci and for some j 6= i, P(aj) =cj and P′(aj) = bj. Also, we don’t know which is the case for which coordinatei. Give an algorithm to find both P and P′. [Hint: First find P + P′ and P · P′.]

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(b) Now, suppose that instead of getting both the values P(ai) and P′(ai) for eachi, we only got one value βi, such that for each i we either have βi = P(ai) orβi = P′(ai). Again, it might happen that for some i, βi = P(ai) while for someother j 6= i, β j = P′(aj) and we don’t know which is the case for which i.However, we are given the promise that

n3≤ |{i ∈ [n] | βi = P(ai)}| ≤

2n3

andn3≤∣∣{i ∈ [n] | βi = P′(ai)

}∣∣ ≤ 2n3

.

Give an algorithm to find both P and P′.

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