We give further results on the weight distributions of the two families of binary codes recently constructed by simplicial complexes by (Wu, Lee, 2020), and show that the converse of the above results is also correct, that is, the binary codes with such weight distributions properties must be these two families of codes. Based on the above results, we also construct another family of binary self-orthogonal codes and present their separating properties and applications to the secret sharing scheme, cryptography and other aspects of information security.

Relative subcodes of a code can be used to describe the security of the un-leaked data symbols in the wire-tap channel with multi-users. In particular, the relative generalized Hamming weight is determined by the minimum support weight of the relative subcodes. We determine the support weight distributions of relative subcodes of several classes of optimal codes which were constructed recently by the down-sets. With respect to certain subcodes with any dimension, the relative generalized Hamming weights of each class of optimal codes are completely determined by using the obtained support weight distributions of relative subcodes. We also show that the relative generalized Hamming weights of each class of optimal codes always achieve the so-called RCW upper bound except few cases.

The symbol-pair simplex codes were introduced by the authors recently, and these codes play a similar role as simplex codes with respect to the Hamming metric. Among other things, the concatenations of the symbol-pair simplex codes are symbol-pair constant-weight codes which are a family of codes achieving the Plotkin-type upper bound of the generalized symbol-pair weight, and thus provide optimal security in the data transmission with the symbol-pair metric in the wire-tap channel with the coset coding scheme. Motivated by the mentioned applications, we present new constructions of symbol-pair simplex codes over any finite field.

We present the security properties of relative one-weight codes when applied to the wire-tap channel, and give the enumeration of equivalent classes of relative one-weight codes. Also, based on the security properties, we define and construct a new class of codes with similar applications as relative one-weight codes, and determine the key parameters of this new class of codes.

We present the Plotkin-type bound on the generalized symbol-pair weight and show that all the symbol-pair equiweight codes achieve the Plotkin-type bound. Some new judging criteria are given for symbol-pair equiweight codes, and based on these judging criteria, we show the existence of a class of symbol-pair equiweight codes by design and graph theory. Also, we explicitly construct the symbol-pair version simplex code by the action of a group on a set. Furthermore, we construct several classes of inequiweight codes achieving the Plotkin-type bound.

Based on the concept of the value assignment and the enumeration techniques in the projective space, we determine the subcode-support-weight distributions for several classes of linear optimal codes which were constructed recently by using the so-called down-sets. Particularly, the generalized Hamming weights of these classes of optimal codes can be determined from their subcode-support-weight distributions. By using the value assignment, we also determine the trellis complexity of each class of optimal codes.

网络舆情治理过程中存在组织碎片、治理过程碎片、决策碎片等情况,造成治理主体信息发布不一致、目标不明确等问题.网络舆情需要各方协同治理.从数据协同、组织协同、全周期协同、监测与决策协同、预案协同、保障协同等方面,探讨了基于大数据技术构建网络舆情协同治理机制的必要性和思路及方法.

Motivated by the concepts of the relative generalized Hamming weight and the greedy weight, the relative greedy weight is introduced, and then it is shown that the codes achieving the upper bounds on the relative greedy weight are optimal on the security of the transmitted data symbols in the wire-tap channel. Based on such applications, the finite geometry method is generalized, and by using the generalized finite geometry method, certain upper bounds on the third relative greedy weight of 4-dimensional codes with respect to 1-dimensional subcodes are first determined, and then optimal codes are constructed with respect to these obtained upper bounds.

The descriptions of generalized Hamming weights with respect to rank are given for codes over chain rings. Based on the descriptions of generalized Hamming weights with respect to rank, the double chain condition, in particular, the chain condition, is introduced and some judging criteria for the double chain condition are presented. As an application of the chain condition, we determine generalized Hamming weights with respect to rank of the tensor product of certain codes satisfying the chain condition. By using generalized Hamming weights with respect to rank, we generalize some results obtained in recent references. We introduce relative generalized Hamming weights to codes over chain rings and principal ideal rings, and present equivalent descriptions and bounds on them. We also generalize maximum distance codes with respect to rank to relative maximum distance codes with respect to rank and give a series of judging criteria for relative maximum distance codes.

The relative greedy weights were introduced in 2017 by Li and Liu. We will solve an open problem therein about the second relative greedy weight of 3-dimensional codes with respect to 1-dimensional subcodes.

We show that a commutative ring which can be applied to the wire-tap channel of type II with the coset coding scheme is a Frobenius one. Relative generalized Hamming weights (RGHWs) are then generalized to codes over Frobenius rings. The basic properties and descriptions of RGHWs for codes over Frobenius rings are presented.

Based on the applications to the wire-tap channel and bounds on the relative generalized Hamming weight, we will introduce optimal codes of type $\mathcal {I}$ and type $\mathcal {II}$ . We will give some necessary conditions and the explicit construction for a linear code $\mathcal {C}$ to be optimal of type $\mathcal {I}$ and $\mathcal {II}$ with respect to a subcode $\mathcal {C}_{1}$ . We will also present a new bound on two different parameters of the relative generalized Hamming weight.

Let R be a commutative Frobenius local ring. A result that the injective hull of an LCD code C over R. is free of dimension 1(C), where 1(C) is the minimum over the cardinalities of the generating sets of C, is proved in this correspondence. Applying this result, a concise proof for the main result in a recent paper by Sanjit Bhowmick et al. is derived. Furthermore, the LCD λ-constacyclic codes with λ being a unit, π(λ 2 ) = 1 and λ 2 ≠ 1, where π is the natural projection of R. to its residue field, are characterized, as another application of our result.

We state that the minimum and the maximum support weight of subcodes of a code are useful in several aspects. Based on the above findings, we will determine part of the above mentioned parameters for two classes of binary codes. We also present the separation property and calculate the trellis complexity of a class of binary codes.

The t-wise intersection of constant-weight codes are computed.Based on the above result,the t-wise intersection of relative two-weight codes are determined by using the finite geometric structure of relative two-weight codes.

Some classes of binary codes constructed by using some defining sets are studied, and for most defining sets, we will determine the generalized Hamming weight of the corresponding codes completely, and for other defining sets, we will determine part of the generalized Hamming weight of the corresponding codes.

The concepts of pseudocodeword and pseudoweight play a fundamental role in the finite-length analysis of LDPC codes. The pseudoredundancy of a binary linear code is defined as the minimum number of rows in a parity-check matrix such that the corresponding minimum pseudoweight equals its minimum Hamming distance. By using the value assignment of Chen and Klove we present new results on the pseudocode-word redundancy of binary linear codes. In particular, we give several upper bounds on the pseudoredundancies of certain codes with repeated and added coordinates and of certain shortened subcodes. We also investigate several kinds of k-dimensional binary codes and compute their exact pseudocodeword redundancy.

By using the cogredience theories of an alternate matrix, a symmetric matrix and a Hermitian-symmetric matrix, we will find a special family of generator matrices for any linear code, and then using the special family of generator matrices, we will provide a general method to construct a linear complementary dual (LCD) code (resp. a Hermitian LCD code) from any given linear code. Still using the special family of generator matrices for LCD codes (resp. Hermitian LCD codes), we will present the enumeration of all [ n , k ] LCD codes (resp. Hermitian LCD codes).

The upper bounds on the difference between the third greedy weight and the third generalized Hamming weight of 4-dimensional q -ary codes are obtained by using the finite geometry method. The codes achieving the upper bounds are constructed, and these codes are optimal with respect to the security when they are used in the wire-tap channel of type II with the coset coding scheme.

By using linear algebra over finite commutative rings, we will present some judging criterions for linear complementary dual (LCD) codes over rings, in particular, free LCD codes over finite commutative rings are described. By using free LCD codes over finite commutative rings and the Chinese Remainder Theorem, LCD codes over semi-simple rings are constructed and the equivalence of free codes and free LCD codes is given. In addition, all the possible LCD codes over chain rings are determined. We also generalize the judging criterion for cyclic LCD codes over finite fields to cyclic LCD codes over chain rings. Based on the above results and the Chinese Remainder Theorem, we also present results for LCD codes over principal ideal rings.