Failures of the MacWilliams Identities for Weights with Maximal Symmetry over Finite Chain Rings and Matrix Rings

Alapfogalmak

The MacWilliams identities, which relate the weight enumerators of a linear code and its dual, often fail to hold for weights with maximal symmetry over finite chain rings and matrix rings over finite fields.

Kivonat

The paper examines the w-weight enumerators of weights w with maximal symmetry over finite chain rings and matrix rings over finite fields. It shows that in many cases, including the homogeneous weight, the MacWilliams identities fail because there exist two linear codes with the same w-weight enumerator whose dual codes have different w-weight enumerators.
The key highlights and insights are:

The MacWilliams identities are known to hold for the Hamming weight enumerator and certain generalizations, but the author shows they often fail for weights with maximal symmetry.

The reason the MacWilliams identities fail is that there exist linear codes C and D with the same w-weight enumerator (wweC = wweD) but different w-weight enumerators of their dual codes (wweC⊥≠ wweD⊥).

For finite chain rings, the author shows that the only weights for which the MacWilliams identities hold are multiples of the Hamming weight and multiples of the homogeneous weight.

For matrix rings over finite fields, the author shows the only weights for which the MacWilliams identities hold are multiples of the Hamming weight (any finite field) and multiples of the homogeneous weight (only for q=2).

More generally, the homogeneous weight on Mk×k(Fq), k≥2, satisfies the MacWilliams identities if and only if k=q=2.

The author develops techniques to construct linear codes with the same w-weight enumerator but different dual code enumerators, as well as to detect differences in the dual code enumerators.

Statisztikák

The paper does not contain any specific numerical data or metrics to support the key arguments.

Idézetek

"The reason the MacWilliams identities often fail is that there exist R-linear codes C, D ⊆Rn for some n such that wweC = wweD but wweC⊥≠ wweD⊥."
"For example, suppose R = Z/4Z and w is an integer-valued weight on R with maximal symmetry (i.e., w(1) = w(3) for this ring). Then the only weights for which the MacWilliams identities hold are multiples of the Hamming weight and multiples of the homogeneous weight (the Lee weight for this ring), Corollary 9.6."
"Similarly, let R = M2×2(Fq) and w be an integer-valued weight with maximal symmetry (i.e., the value of w(r) depends only on the rank of r). Then the only weights for which the MacWilliams identities hold are multiples of the Hamming weight (any q) and multiples of the homogeneous weight (q = 2 only), Theorem 19.5."

Főbb Kivonatok

Weights with Maximal Symmetry and Failures of the MacWilliams Identities

by Jay A. Wood : arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.07154.pdf

Mélyebb kérdések

What are some potential applications or implications of the failure of the MacWilliams identities for weights with maximal symmetry

The failure of the MacWilliams identities for weights with maximal symmetry has several potential applications and implications. One significant implication is in the realm of error-correcting codes. The MacWilliams identities play a crucial role in understanding the relationship between the weight enumerators of a linear code and its dual code. When these identities fail for weights with maximal symmetry, it can lead to the discovery of new types of codes or coding schemes that may have unique properties or advantages. This could potentially open up avenues for developing more efficient or robust error-correcting codes in communication systems.
Another implication could be in the field of algebraic coding theory. The failures of the MacWilliams identities can provide insights into the algebraic structures of finite chain rings and matrix rings over finite fields. Understanding these failures can lead to new theoretical developments and applications in algebraic coding theory, potentially leading to advancements in coding techniques and algorithms.
Furthermore, the failures of the MacWilliams identities can also have implications in cryptography and data security. By exploring the reasons behind these failures and studying the properties of weights with maximal symmetry, researchers may uncover new cryptographic techniques or encryption methods that leverage these unique characteristics for enhanced security and privacy protection.

Can the techniques developed in this paper be extended to study the MacWilliams identities for other types of weights or algebraic structures beyond finite chain rings and matrix rings over finite fields

The techniques developed in the paper can be extended to study the MacWilliams identities for other types of weights or algebraic structures beyond finite chain rings and matrix rings over finite fields. One possible extension could involve investigating the MacWilliams identities for weights with maximal symmetry in different algebraic structures such as finite groups, modules, or other types of rings. By adapting the methods and concepts used in the paper to these new settings, researchers can explore the behavior of weight enumerators and their relationships in diverse algebraic contexts.
Additionally, the techniques developed in the paper could be applied to study the MacWilliams identities for weights with specific properties or symmetries in various mathematical structures. By generalizing the concepts and results obtained in the paper, researchers can explore the applicability of the MacWilliams identities in different mathematical frameworks and investigate the conditions under which these identities hold or fail.
Overall, the techniques presented in the paper provide a foundation for studying weight enumerators and their connections to dual codes in various algebraic structures, paving the way for further research and exploration in the field of coding theory and algebraic coding.

Are there any connections between the failure of the MacWilliams identities and the structure or properties of the underlying finite rings

There are indeed connections between the failure of the MacWilliams identities and the structure or properties of the underlying finite rings. The properties of the finite chain rings and matrix rings over finite fields, such as the existence of a unique maximal left ideal and the chain structure of ideals, play a crucial role in determining the behavior of weight enumerators and the validity of the MacWilliams identities.
The failure of the MacWilliams identities in these specific algebraic structures can be attributed to the symmetries and properties of the weights defined on these rings. For weights with maximal symmetry, the interactions between the weight function and the ring structure can lead to discrepancies in the weight enumerators of linear codes and their dual codes, resulting in the failures of the MacWilliams identities.
Furthermore, the specific characteristics of the weight functions, such as their values and symmetries, interact with the algebraic properties of the rings to influence the behavior of the weight enumerators. Understanding these connections can provide insights into the interplay between algebraic structures and weight functions in the context of coding theory and algebraic coding, shedding light on the complexities of error-correcting codes and their properties in diverse mathematical settings.