## 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 *A*_{q}(n,d) denote the maximum M such that there exists a q-ary (n, M, d)-code.

**Prob. 1**: Show that A_{q}(n,1) = q^{n}.

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

72 A_{2}(10,3) 79 and

144 A_{2}(11,3) 158.

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

**Prob. 2**: Prove that A_{2}(3,2) = 4.

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

(This is a* sphere-packing upper bound *on A_{q}(n,d). )

**Prob. 4:** For arbitrary d show that

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

**Prob. 5:** Show that 19 A_{2}(10,3) 93.

**Prob. 6:** Prove that A_{2}(5,3) = 4 and show that there is a unique (up to equivalence) binary
(5,4,3) -code.

**Prob. 7:** Prove that A_{2}(n,d) 2A_{2}(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 A_{q}(n,d) q^{n - 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 A_{2}(n,d) 2^{k}, where k is the largest integer satisfying the above
inequality. (This is the *Gilbert-Varshamov bound* in the case q = 2).

Return to M5410 Homepage