On Some Linear Codes over Z_2^s

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On Some Linear Codes over Z_2^s

Author(s):

Creation Date: Dec 11, 1999

Published In: Jul 2000

Paper Type: Dissertation

Address: Department of Mathematics & Statistics, IIT Kanpur, India

School: Indian Institute of Technology Kanpur India

Abstract:

Algebraic Coding Theory (ACT) is a branch of engineering with roots in mathematics and applications to computer science. Its application ranges from deep-space communications to consumer electronics. Codes over ring of integers modulo $m$ is a recent branch of ACT with applications including phase modulated channels, multifrequency phase telegraphy, multilevel quantized pulse amplitude modulated channels, orthogonal frequency division multiplexing (OFDM) and Code Division Multiple Access (CDMA) communication. In my dissertation I have studied various linear codes over the ring of integers modulo $2^{s}$ and obtained their fundamental properties like, 2-dimension, Hamming weight distributions, Lee weight distributions, and Generalized Hamming weights (GHW) etc. The following industrial applications of GHW's are well known, e.g. wire-tap channel-II, t-resilient functions, trellis coding (lower bounding the number of trellis states) and in truncating a linear block code. Most of the communication engineers are familiar with the single error-correcting binary Hamming Code because of its various applications. I have given analogue of these over the ring of integers modulo $2^{s}$ and obtained from them some interesting binary codes (uniformly packed) via Gray images. I have been able to extend a concept of chain condition which was earlier used for codes over fields to obtain the weight hierarchies of product codes to codes over rings and have shown that various known codes over ring satisfy the chain condition.


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