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On p-adic estimates of weights in Abelian codes over Galois rings

Citation

Katz, Daniel J. (2005) On p-adic estimates of weights in Abelian codes over Galois rings. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-05312005-175744

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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Ax-Katz; Chevalley-Warning; cyclic codes; Delsarte; error-correcting codes; McEliece; p-divisibility; polynomials
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Wilson, Richard M.
Thesis Committee:
  • Wilson, Richard M. (chair)
  • Wales, David B.
  • Ramakrishnan, Dinakar
  • McEliece, Robert J.
Defense Date:11 May 2005
Author Email:katz (AT) its.caltech.edu
Record Number:CaltechETD:etd-05312005-175744
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-05312005-175744
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2329
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:01 Jun 2005
Last Modified:26 Dec 2012 02:50

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