In this paper, we give a notation on the Singleton bounds for linear codes over a finite commutative quasi-Frobenius ring in the work of Shiromoto [5]. We show that there exists a class of finite commutative quasi-Frobenius rings. The Singleton bounds for linear codes over such rings satisfy d(C) − 1 A ≤ n− log|R| |C|. keywords : Linear codes, General weight, Quasi-Frobenius ring, Singleton bounds

Recall that a finite-dimensional commutative associative algebra equipped with an invariant nondegenerate symmetric bilinear form is called a Frobenius algebra (here, we do not require an existence of a unit in Frobenius algebra). Any commutative quasi-Frobenius algebra is always Frobenius, i.e., if the identity ab = ba (commutativity) is fulfilled in a quasi-Frobenius algebra, then the identities

let r be a commutative ring with non-zero identity and m be a unital r-module. then the concept of quasi-secondary submodules of m is introduced and some results concerning this class of submodules is obtained

We study quasi-Hopf algebras and their subobjects over certain commutative rings from the point of view of Frobenius algebras. We introduce a type of Radford formula involving an anti-automorphism and the Nakayama automorphism of a Frobenius algebra, then view several results in quantum algebras from this vantage-point. In addition, separability and strong separability of quasi-Hopf algebras ar...

A zero-knowledge authentication scheme is a type of authentication scheme, which gives no knowledge beyond the authenticity identifying an entity and is probabilistic than deterministic authentication scheme. This paper proposes Diffie-Hellman and Fiat-Shamir like zero-knowledge authentication schemes on general non-commutative groups. The key idea of our proposal is that for a given non-commut...

We extend our previous work on the quasi-particle excitations in N = 4 non-commutative U(1) Yang-Mills theory at finite temperature. We show that above some critical temperature there is a tachyon in the spectrum of excitations. It is a collective transverse photon mode polarized in the non-commutative plane. Thus the theory seems to undergo a phase transition at high temperature. Furthermore w...

In this brief note we would like to give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product. —————————————

We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula. Résumé. Nous introduisons une nouvelle base des fonctions symé...

In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform sol...