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
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|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||11 May 2005|
|Non-Caltech Author Email:||katz (AT) its.caltech.edu|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||01 Jun 2005|
|Last Modified:||26 Dec 2012 02:50|
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