Properties of codes with two homogeneous weights

Titelangaben

Byrne, Eimear ; Kiermaier, Michael ; Sneyd, Alison:
Properties of codes with two homogeneous weights.
In: Finite Fields and their Applications. Bd. 18 (2012) Heft 4 . - S. 711-727.
ISSN 1071-5797
DOI: https://doi.org/10.1016/j.ffa.2012.01.002

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Abstract

Delsarte showed that for any projective linear code over a finite field GF(pʳ) with two nonzero Hamming weights w₁ < w₂ there exist positive integers u and s such that w₁ = pˢu and w₂ = pˢ(u+1). Moreover, he showed that the additive group of such a code has a strongly regular Cayley graph. Here we show that for any regular projective linear code C over a finite Frobenius ring with two integral nonzero homogeneous weights w₁ < w₂ there is a positive integer d, a divisor of |C|, and positive integer u such that w₁ = du and w₂ = d(u+1). This gives a new proof of the known result that any such code yields a strongly regular graph. We apply these results to existence questions on two-weight codes.