Citation
Katz, Daniel Jerome (2005) On pAdic Estimates of Weights in Abelian Codes over Galois Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6NSP2A36. https://resolver.caltech.edu/CaltechETD:etd05312005175744
Abstract
Let p be a prime. We prove various analogues and generalizations of McEliece's theorem on the pdivisibility 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 padic 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^{d}Z. The first part gives a lower bound on the padic 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^{d}Z in words of our code; we call this number the scount. We find a j such that p^{j} divides the scounts 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 2adic 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 padic valuations of Hamming weights. When we specialize this result to finite fields, we recover the theorem of Delsarte and McEliece on the pdivisibility of weights in Abelian codes over finite fields.
The fourth result generalizes the DelsarteMcEliece 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 padically 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 pdivisibility of the cardinalities of affine algebraic sets over finite fields.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  AxKatz; ChevalleyWarning; cyclic codes; Delsarte; errorcorrecting codes; McEliece; pdivisibility; polynomials 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Awards:  Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2001, 2003. 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  11 May 2005 
NonCaltech Author Email:  daniel.katz (AT) csun.edu 
Record Number:  CaltechETD:etd05312005175744 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd05312005175744 
DOI:  10.7907/6NSP2A36 
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 ETDdb 
Deposited On:  01 Jun 2005 
Last Modified:  22 May 2020 20:28 
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