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On a class of constant weight codes. (English) Zbl 0860.94030
Electron. J. Comb. 3, No. 1, Research paper R4, 13 p. (1996); printed version J. Comb. 3, No. 1, 43-55 (1996).
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.
MSC:
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11T23 Exponential sums