## 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 A2(10,3) 79 and
144 A2(11,3) 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

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

Prob. 4: For arbitrary d show that

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

Prob. 5: Show that 19 A2(10,3) 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) 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) qn - d + 1. (For linear codes this is known as the Singelton bound).

Prob 10: Show that if

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

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