Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

Diffusion-weighted imaging of the brain at 7 T with echo-planar and turbo spin echo sequences: preliminary results

Ultra-high-field clinical MRI scanners (e.g., 7 T and above) are becoming increasingly prevalent and can potentially enhance diagnostic ability through higher contrast, resolution and/or sensitivity. Diffusion-weighted MRI is a highly valued component in today's radiological exam and may benefit from the enhanced signal-to-noise ratio provided by high field with the appropriate imaging strategy. The most common diffusion pulse sequence readout (echo-planar imaging (EPI)) has been widely employed for in vivo human 7 T diffusion tensor imaging (DTI). In this article, we present results of brain DTI at 7 T with two diffusion-weighted imaging pulse sequence readouts: echo-planar imaging (EPI-DTI) and turbo spin echo (TSE-DTI). Results indicate that analogous coverage, quality and resolution typical of lower field (2 mm) can be obtained by properly processed EPI-DTI at 7 T, and, with some reduction in efficiency and sharpness, TSE-DTI at 7 T. Furthermore, 7 T TSE-DTI shows promise in obtaining higher-resolution results in targeted acquisitions of specific brain areas.     

Improved Bounds for Ternary Linear Codes of Dimension 8 Using Tabu Search

In this paper, new codes of dimension 8 are presented which give improved bounds on the maximum possible minimum distance of ternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. Twenty three codes are given which improve or establish the bounds for ternary codes. In addition, a table of upper and lower bounds for d ₃(n, 8) is presented for n 200.     

Optimal Ternary Linear Rate 1/2 Codes

In this paper, we classify all optimal linear[n, n/2] codes up to length 12. We show that thereis a unique optimal [10, 5, 5] code up to equivalence.     

Generalized Steiner systems GS4 (2, 4, v,g) for g = 2, 3, 6

Generalized Steiner systems GS_d(t, k, v, g) were first introduced by Etzion and used to construct optimal constant‐weight codes over an alphabet of size g + 1 with minimum Hamming distance d, in which each codeword has length v and weight k. Much work has been done for the existence of generalized Steiner triple systems GS(2, 3, v, g). However, for block size four there is not much known on GS_d(2, 4, v, g). In this paper, the necessary conditions for the existence of a GS_d(t, k, v, g) are given, which answers an open problem of Etzion. Some singular indirect product constructions for GS_d(2, k, v, g) are also presented. By using both recursive and direct constructions, it is proved that the necessary conditions for the existence of a GS₄(2, 4, v, g) are also sufficient for g = 2, 3, 6. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 401–423, 2001     

Structure Theorems for Group Ring Codes with an Application to Self-Dual Codes

Using ideas from the cohomology of finite groups, an isomorphism is established between a group ring and the direct sum of twisted group rings. This gives a decomposition of a group ring code into twisted group ring codes. In the abelian case the twisted group ring codes are (multi-dimensional) constacyclic codes. We use the decomposition to prove that, with respect to the Euclidean inner product, there are no self-dual group ring codes when the group is the direct product of a 2-group and a group of odd order, and the ring is a field of odd characteristic or a certain modular ring. In particular, there are no self-dual abelian codes over the rings indicated. Extensions of these results to non-Euclidean inner products are briefly discussed.     

Rankin-Cohen Brackets and Invariant Theory

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.     

Self-reciprocal and self-conjugate-reciprocal irreducible factors of xn − λ and their applications

In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of $x^n-\lambda$, $\lambda \in {\mathbb{F}}_q^{⁎}$, over ${\mathbb{F}}_q$, which are just open questions posed by Boripan et al. (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.