On p-Adic Estimates of Weights in Abelian Codes over Galois Rings

• Main Author: Katz, Daniel Jerome
• Format: Others
• Published: 2005
• Online Access: https://thesis.library.caltech.edu/2329/1/djkatz_thesis.pdf; Katz, Daniel Jerome (2005) On p-Adic Estimates of Weights in Abelian Codes over Galois Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6NSP-2A36. https://resolver.caltech.edu/CaltechETD:etd-05312005-175744 <https://resolver.caltech.edu/CaltechETD:etd-05312005-175744>

<p>Let p be a prime. We prove various analogues and generalizations of McEliece's theorem on the p-divisibility of weights of words in cyclic codes over a finite field of characteristic p. Here we consider Abelian codes over various Galois rings. We present four new theorems on p-adic valuations of weights. For simplicity of presentation here, we assume that our codes do not contain constant words.</p> <p>The first result has two parts, both concerning Abelian codes over Z/p^dZ. The first part gives a lower bound on the p-adic valuations of Hamming weights. This bound is shown to be sharp: for each code, we find the maximum k such that p^k divides all Hamming weights. The second part of our result concerns the number of occurrences of a given nonzero symbol s ∈ Z/p^dZ in words of our code; we call this number the s-count. We find a j such that p^j divides the s-counts of all words in the code. Both our bounds are stronger than previous ones for infinitely many codes.</p> <p>The second result concerns Abelian codes over Z/4Z. We give a sharp lower bound on the 2-adic valuations of Lee weights. It improves previous bounds for infinitely many codes.</p> <p>The third result concerns Abelian codes over arbitrary Galois rings. We give a lower bound on the p-adic valuations of Hamming weights. When we specialize this result to finite fields, we recover the theorem of Delsarte and McEliece on the p-divisibility of weights in Abelian codes over finite fields.</p> <p>The fourth result generalizes the Delsarte-McEliece theorem. We consider the number of components in which a collection c_1,...,c_t of words all have the zero symbol; we call this the simultaneous zero count. Our generalized theorem p-adically estimates simultaneous zero counts in Abelian codes over finite fields, and we can use it to prove the theorem of N. M. Katz on the p-divisibility of the cardinalities of affine algebraic sets over finite fields.</p>