Negacyclic codes of prime power length over the finite non-commutative chain ring 𝔽pm[u,𝜃] 〈u2〉

In this paper, we focus on the algebraic structure of left negacyclic codes of length [Formula: see text] over the finite non-commutative chain ring [Formula: see text] where [Formula: see text] is an automorphism on [Formula: see text]. After that, the number of codewords of all left negacyclic codes is obtained. For each left negacyclic code, we also obtain the structure of its right dual code. In the remaining result, the number of distinct left negacyclic codes is given. Finally, a one-to-one correspondence between left negacyclic and left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] is constructed via ring isomorphism, which carries over the results regarding left negacyclic codes corresponding to left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] is a nonzero element of the field [Formula: see text] such that [Formula: see text].

Decoding Algorithm by Cooperation Between Hartmann Rudolph Algorithm and a Decoder Based on Syndrome and Hash

In this paper, the authors present a concatenation of Hartmann and Rudolph (HR) partially exploited and a decoder based on hash techniques and syndrome calculation to decode linear block codes. This work consists firstly to use the HR with a reduced number of codewords of the dual code then the HWDec which exploits the output of the HR partially exploited. Researchers have applied the proposed decoder to decode some Bose, Chaudhuri, and Hocquenghem (BCH) and quadratic residue (QR) codes. The simulation and comparison results show that the proposed decoder guarantees very good performances, compared to several competitors, with a much-reduced number of codewords of the dual code. For example, for the BCH(31, 16, 7) code, the good results found are based only on 3.66% of the all codewords of the dual code space, for the same code the reduction rate of the run time varies between 78% and 90% comparing to the use of Hartmann and Rudolph alone. This shows the efficiency, the rapidity, and the reduction of the memory space necessary for the proposed concatenation.

Research on Chinese word-formation

Pinyin text uses letters that are closely related to speech to record speech, but we don’t know much about how ideographic text is represented. We analyzed the relationship between the glyph and the meaning of nine thousand Chinese characters and found that the combination of the glyph is actually to construct imagery, and this imagery is the psychological representation of the experience situation that forms the concept of its character, so the meaning and glyph of the characters are ideographically connected. We find out the imagery and system structure of these 9,000 Chinese characters, and summarize the five cognitive modes of their combination. These results reveal the ideographic mechanism of the dual-code hierarchical combination of the internal formation of Chinese characters and its imagery, demonstrate the application of cognitive linguistics and cognitive psychology in the study of ideographic characters, and put forward a guiding theory of Chinese word-formation.

A Study on Multisecret-Sharing Schemes Based on Linear Codes

Secret sharing has been a subject of study since 1979. In the secret sharing schemes there are some participants and a dealer. The dealer chooses a secret. The main principle is to distribute a secret amongst a group of participants. Each of whom is called a share of the secret. The secret can be retrieved by participants. Clearly the participants combine their shares to reach the secret. One of the secret sharing schemes is threshold secret sharing scheme. A threshold secret sharing scheme is a method of distribution of information among participants such that can recover the secret but cannot. The coding theory has been an important role in the constructing of the secret sharing schemes. Since the code of a symmetric design is a linear code, this study is about the multisecret-sharing schemes based on the dual code of code of a symmetric design. We construct a multisecret-sharing scheme Blakley’s construction of secret sharing schemes using the binary codes of the symmetric design. Our scheme is a threshold secret sharing scheme. The access structure of the scheme has been described and shows its connection to the dual code. Furthermore, the number of minimal access elements has been formulated under certain conditions. We explain the security of this scheme.

Improved Decoding of Linear Block Codes by Concatenated Decoders

The use of decoding algorithms allows us to retrieve information after transmitting it over a noisy communication channel. Soft decision decoding is powerful in concatenation schemes that use two or more levels of decoding. In our case, we make a concatenation between the Hartmann & Rudolph (HR) algorithm as symbol-by-symbol decoder and the chase-2 algorithm that is word-to-word decoding algorithm. In this paper, we propose to combine two decoding algorithms for constructing a new one with more efficiency and less complexity. This work consists firstly to use the HR with a reduced number of codewords of the dual code then the Chase-2 algorithm which exploits the output of PHR. The simulations results and the comparisons made show that the proposed decoding scheme guarantees very good performance with reduced temporal complexity.

DUAL CODE AND MACWILLIAMS IDENTITY ON ADDITIVE CODE

Additive code is a generalization of linear code. It is defined as subgroup of a finite Abelian group. The definitions of Hamming distance, Hamming weight, weight distribution, and homogeneous weight distribution in additive code are similar with the definitions in linear code. Different with linear code where the dual code is defined using inner product, additive code using theories in group to define its dual code because in group theory we do not have term of inner product. So, by this thesis, the definitions of dual code in additive code will be discussed. Then, this thesis discuss about a familiar theorem in dual code theory, that is MacWilliams Identity. Next, this thesis discuss about how to proof of MacWilliams Identity on adiitive code using dual codes which are defined.

Extended maximal self-orthogonal codes

Self-dual and maximal self-orthogonal codes over [Formula: see text], where [Formula: see text] is even or [Formula: see text](mod 4), are extensively studied in this paper. We prove that every maximal self-orthogonal code can be extended to a self-dual code as in the case of binary Golay code. Using these results, we show that a self-dual code can also be constructed by gluing theory even if the sum of the lengths of the gluing components is odd. Furthermore, the (Hamming) weight enumerator [Formula: see text] of a self-dual code [Formula: see text] is given in terms of a maximal self-orthogonal code [Formula: see text], where [Formula: see text] is obtained by the extension of [Formula: see text].

Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4)

Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22.