Math 4410/5410 Homework Assignment # 8

Note: The following definitions and problems apply to any code, not just linear ones. When linear codes are discussed they will be explicitly identified.

Def: If a code has M codewords of length n and has minimum distance d, then it is called an (n, M, d)-code.

Def: Let Aq(n,d) denote the maximum M such that there exists a q-ary (n, M, d)-code.

Prob. 1: Show that Aq(n,1) = qn.

In general, the value of Aq(n,d) is not known even for small values of the parameters. For instance, at present we know only that:

72 less than or equal A2(10,3) less than or equal 79 and
144 less than or equal A2(11,3) less than or equal 158.

To cite just two examples where the value has not been determined.

Prob. 2: Prove that A2(3,2) = 4.

Prob. 3: Let d = 2e + 1. Prove that

[A_q(n,d) Sum from {k=0} to e C(n,k) (q-1)^k less than or equal to q^n]

(This is a sphere-packing upper bound on Aq(n,d). )

Prob. 4: For arbitrary d show that

[A_q(n,d) Sum from {i=0} to {d-1} C(n,i) (q-1)^i greater than or equal to q^n]

(This is known as the Gilbert-Varshamov lower bound).

Prob. 5: Show that 19 less than or equal A2(10,3)less than or equal 93.

Prob. 6: Prove that A2(5,3) = 4 and show that there is a unique (up to equivalence) binary (5,4,3) -code.

Prob. 7: Prove that A2(n,d) less than or equal 2A2(n-1,d).

Prob. 8: Prove that over a binary alphabet, if there exists an (n, M, 2k)-code then there exists an (n, M, 2k)-code with all codewords of even weight.

Prob. 9: Prove that Aq(n,d) less than or equal qn - d + 1. (For linear codes this is known as the Singelton bound).

Prob 10: Show that if

[2^k Sum from {i=0} to {d-2} C(n-1,i) (q-1)^i less than 2^n]

then there exists a binary linear [n,k]-code with minimum distance at least d.

Deduce from this that A2(n,d) greater than or equal 2k, where k is the largest integer satisfying the above inequality. (This is the Gilbert-Varshamov bound in the case q = 2).

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