In this correspondence, new extremal self-dual codes of length 62 are constructed with weight enumerators of three different types. Two of these types were not represented by any known code up till now. All these codes possess an automorphism of order 15. Some of them are used to construct extremal self-dual codes of length 60 by the method of subtracting. Manuscript received November 20, 2001;...

The Kneser-Hecke-operator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of self-dual codes of fixed length. It maps a linear self-dual code C over a finite field to the formal sum of the equivalence classes of those self-dual codes that intersect C in a codimension 1 subspace. The eigenspaces of this self-adjoint linear operator may be d...

Self-dual maximum distance separable codes (self-dual MDS codes) and self-dual near (NMDS) are very important in coding theory practice. Thus, it is interesting to construct or NMDS codes. In this paper, we not only give parity-check matrices of dual twisted generalized Reed–Solomon (TGRS but also present a sufficient necessary condition TGRS over $${\mathbb {F}}_q$$ with q an odd prime. Moreov...

The applicability of techniques in coding theory to problems in cryptography is illustrated by examples concerning secret-sharing schemes with tailored access priveleges, the design of perfect local randomizers, the construction of t-resilient functions, and the quantization of the nonlinearity of boolean functions. Some novel coding concepts, in particular the notions of minimal codewords in l...

Self-dual codes have been received great attention by researchers since the beginning of the coding theory. In this work, some construction methods for this kind of codes are composed which produce new self-dual codes from self-dual codes of smaller lengths. A special one of these methods that is called recursive algorithm is also mentioned. For the binary case, it was shown that recursive algo...

This mixed method study compared mathematics achievement in two third-grade classrooms using two different representations: virtual manipulatives which has dual codes of visual and symbolic representations, and physical manipulatives for instruction in fraction addition and balancing algebraic equations. The research employed a within-subjects crossover repeated measures design, administered pr...

In this short note, we describe one of the long-standing open problems in algebraic coding theory, i.e., whether there exists a binary self-dual [72, 36, 16] code.

Based on dual coding theory, this study predicted that pictorial typography (PT), a device involving direct visual integration of a written word into a picture of its referent, is more effective for ESL children’s pre-literacy skills, than the non-integrated paring of word and picture on flashcards (FC). The two types of instruction were given to thirty-two Korean children aged six and seven, b...

In this paper, we address optimal network coding schemes for two-user unicast networks. Using previous results on the relationship between graph structure and informationtheoretic capacity, we characterize the capacity region for a class of two-user unicast networks by finding a combination network coding/routing scheme to achieve an outer bound on capacity. This scheme has the remarkable prope...