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Type of Document	Dissertation
Author	Katz, Daniel J.
Author's Email Address	katz AT its.caltech.edu
URN	etd-05312005-175744
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-05312005-175744
Title	On p-adic estimates of weights in Abelian codes over Galois rings
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title R.M. Wilson Committee Chair David B. Wales Committee Member Dinakar Ramakrishnan Committee Member Robert J. McEliece Committee Member
Keywords	Chevalley-Warning Delsarte McEliece cyclic codes error-correcting codes Ax-Katz polynomials p-divisibility
Date of Defense	2005-05-11
Availability	unrestricted
Abstract 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. 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 in 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. 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. 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. 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.
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