## 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
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