On a class of constant weight codes (Q1918864)

Summary: For any odd prime power \(q\) we first construct a certain nonlinear binary code \(C(q,2)\) having \((q^2-q)/2\) codewords of length \(q\) and weight \((q-1)/2\) each, for which the Hamming distance between any two distinct codewords is in the range \([q/2- 3\sqrt q/2,\;q/2+3\sqrt q/2]\), that is, `almost constant'. Moreover, we prove that \(C(q,2)\) is distance-invariant. Several variations and improvements on this theme are then pursued. Thus, we produce other classes of binary codes \(C(q,n)\), \(q\geq 3\), of length \(q\) that have `almost constant' weights and distances, and which, for fixed \(n\) and big \(q\), have asymptotically \(q^n/n\) codewords. Then we prove the possibility of extending our codes by adding the complements of their codewords. Also, by using results on Artin \(L\)-series, it is shown that the distribution of the 0's and 1's in the codewords we constructed is quasi-random. Our construction uses character sums associated with the quadratic character \(\chi\) of \(F_{q^n}\) in which the range of summation is \(F_q\). Relations with the duals of the double error correcting BCH codes and the duals of the Melas codes are also discussed.