scholarly journals MacWilliams Identities and Matroid Polynomials

10.37236/1636 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Thomas Britz
Keyword(s):  
Linear Code ◽  
Tutte Polynomial ◽  
Decomposition Theorem ◽  
Weight Enumerators ◽  
Support Weight ◽  
Macwilliams Identities

We present generalisations of several MacWilliams type identities, including those by Kløve and Shiromoto, and of the theorems of Greene and Barg that describe how the Tutte polynomial of the vector matroid of a linear code determines the $r$th support weight enumerators of the code. One of our main tools is a generalisation of a decomposition theorem due to Brylawski.

MACWILLIAMS IDENTITIES FOR WEIGHT ENUMERATORS WITH RESPECT TO THE RT METRIC

2014 ◽  
Vol 06 (02) ◽  
pp. 1450030 ◽  
Author(s):  
AMIT K. SHARMA ◽  
ANURADHA SHARMA
Keyword(s):  
Algebraic Structure ◽  
Linear Code ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Weight Enumerator ◽  
Weight Enumerators ◽  
Ring Of Integers ◽  
The Past ◽  
Macwilliams Identities ◽  
Hamming Metric

Linear codes constitute an important family of error-correcting codes and have a rich algebraic structure. Initially, these codes were studied with respect to the Hamming metric; while for the past few years, they are also studied with respect to a non-Hamming metric, known as the Rosenbloom–Tsfasman metric (also known as RT metric or ρ metric). In this paper, we introduce and study the split ρ weight enumerator of a linear code in the R-module Mn×s(R) of all n × s matrices over R, where R is a finite Frobenius commutative ring with unity. We also define the Lee complete ρ weight enumerator of a linear code in Mn×s(ℤk), where ℤk is the ring of integers modulo k ≥ 2. We also derive the MacWilliams identities for each of these ρ weight enumerators.

A generalization of the Whitney rank generating function

1993 ◽  
Vol 113 (2) ◽  
pp. 267-280 ◽  
Author(s):  
G. E. Farr
Keyword(s):  
Generating Function ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Natural Generalization ◽  
Chromatic Polynomial ◽  
Weight Enumerator ◽  
Hadamard Transform ◽  
Percolation Probability ◽  
Special Cases

AbstractThe Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.

NOTE ON SUPPORT WEIGHT DISTRIBUTION OF LINEAR CODES OVER

2015 ◽  
Vol 91 (2) ◽  
pp. 345-350 ◽  
Author(s):  
JIAN GAO
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Linear Codes ◽  
Dual Code ◽  
Support Weight

AbstractLet $R=\mathbb{F}_{p}+u\mathbb{F}_{p}$, where $u^{2}=u$. A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{2k}$ over $R$ and its dual code $\mathscr{C}^{\bot }$ is established.

scholarly journals Cycle Index, Weight Enumerator, and Tutte Polynomial

10.37236/1663 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Peter J. Cameron
Keyword(s):  
Permutation Group ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Weight Enumerator ◽  
Permutation Groups ◽  
Cycle Index ◽  
Transitive Groups ◽  
Index Weight

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arises directly from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene's Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa. I propose a polynomial for IBIS groups which generalises both Tutte polynomial and cycle index.

The third support weight enumerators of the doubly-even, self-dual [32, 16, 8] codes

2003 ◽  
Vol 49 (3) ◽  
pp. 740-746 ◽  
Author(s):  
O. Milenkovic ◽  
S.T. Coffey ◽  
K.J. Compton
Keyword(s):  
Weight Enumerators ◽  
Support Weight ◽  
The Third

ON THE SUPPORT WEIGHT DISTRIBUTION OF LINEAR CODES OVER THE RING

2016 ◽  
Vol 95 (1) ◽  
pp. 157-163 ◽  
Author(s):  
MINJIA SHI ◽  
JIAQI FENG ◽  
JIAN GAO ◽  
ADEL ALAHMADI ◽  
PATRICK SOLÉ
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Linear Codes ◽  
Dual Code ◽  
Support Weight

Let $R=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}+\cdots +u^{d-1}\mathbb{F}_{p}$, where $u^{d}=u$ and $p$ is a prime with $d-1$ dividing $p-1$. A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{dk}$ over $R$ and the dual code $\mathscr{C}^{\bot }$ is established.

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