On triply even binary codes

A triply even code is a binary linear code in which the weight of every codeword is divisible by 8. We show how two doubly even codes of lengths m_1 and m_2 can be combined to make a triply even code of length m_1+m_2, and then prove that every maximal triply even code of length 48 can be obtained by combining two doubly even codes of length 24 in a certain way. Using this result, we show that there are exactly 10 maximal triply even codes of length 48 up to equivalence.Comment: 21 pages + appendix of 10 pages. Minor revisio

Classification of Δ\Delta-divisible linear codes spanned by codewords of weight Δ\Delta

We classify all $q$-ary $\Delta$-divisible linear codes which are spanned by codewords of weight $\Delta$. The basic building blocks are the simplex codes, and for $q=2$ additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight $4$ have been classified, which is the case $q=2$ and $\Delta=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $\Delta$-divisible codes of length $4\Delta$ in the projective case.Comment: 12 page

On Divisible Codes over Finite Fields

We study a certain kind of linear codes, namely divisible codes, over finite fields. These codes, introduced by Harold N. Ward, have the property that all codeword weights share a common divisor larger than 1. These are interesting error-correcting codes because many optimal codes and/or classical codes exhibit nontrivial divisibility. We first introduce an upper bound on dimensions of divisible codes in terms of their weight spectrums, as well as a divisibility criteria for linear codes over arbitrary finite fields. Both the bound and the criteria are given by Ward, and these are the primary results that initiate this work. Our first result proves an equivalent condition of Ward's bound, which involves only some property of the weight distribution, but not any other properties (including the linearity) of the code. This equivalent condition consequently provides an alternative (and more elementary) proof of Ward's bound, and from the equivalence we extend Ward's bound to certain nonlinear codes. Another perspective of the equivalence gives rise to our second result, which studies weights modulo a prime power in divisible codes. This is generalized from weights modulo a prime power in linear codes, and yields much better results than the linear code version does. With a similar method we propound a new bound that is proved to be better than Ward's bound. Our third result concerns binary divisible codes of maximum dimension with given lengths. We start with level one and level two codes, which are well described from this point of view. For higher level codes we prove an induction theorem by using the binary version of the divisibility criteria, as well as Ward's bound and the new generated bound. Moreover, this induction theorem allows us to determine the exact bound and the codes that attain the bound for level three codes of relatively small length.</p