Asymptotically good families of subfield subcodes of geometric Goppa codes

We show that subfield subcodes of certain geometric Goppa codes meet the Gilbert-Varshamov bound. A very special case of our theorem is the well known fact that classical Goppa codes meet the Gilbert-Varshamov bound.     

一类代数几何码的构造

利用有限域Fq^8(s≥1为正整数，q为素数幂)上代数曲线构造了一类q元线性码，这类线性码是q^8元几何Goppa码的子域子码的子码，同时也是Chaoping Xing，SanLing构造的代数几何码[1]的推广。     

The lengths of Hermitian self-dual extended duadic codes

Duadic codes are a class of cyclic codes that generalize quadratic residue codes from prime to composite lengths. For every prime power q, we characterize integers n such that there is a duadic code of length n over F_q² with a Hermitian self-dual parity-check extension. We derive asymptotic estimates for the number of such n as well as for the number of lengths for which duadic codes exist.     

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .     

The Galois variance of constacyclic codes

This paper studies the Galois images of constacyclic codes over ${\mathbb{F}}_{q^m}$ of length relatively prime to q, and determines when those images are equal and when they intersect only at the zero codeword. The subfield subcodes and trace codes of constacyclic codes are also determined.     

Subcodes of the Projective Generalized Reed-Muller Codes Spanned by Minimum-Weight Vectors

We use methods of Mortimer [19] to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals. These methods provide a proof, alternative to a dimension argument, that neither the projective generalized Reed-Muller code of order r and of length over the finite field F _q of prime-power order q, nor its dual, is spanned by its minimum-weight vectors for 0<r<m–1 unless q is prime. The methods of proof are the projective analogue of those developed in [17], and show that the codes spanned by the minimum-weight vectors are spanned over F _q by monomial functions in the m variables. We examine the same question for the subfield subcodes and their duals, and make a conjecture for the generators of the dual of the binary subfield subcode when the order r of the code is 1.     

On the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups

There are seven binary extremal self-dual doubly-even codes which are known to have a 2-transitive automorphism group. Using representation theoretical methods we show that there are no other such codes, except possibly for length n = 1024. We also classify all extremal ternary self-dual and quaternary Hermitian self-dual codes.     

Directed graph representation of half-rate additive codes over GF(4)

We show that (n, 2 ⁿ ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 ⁿ ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.     

Galois invariance,trace codes and subfield subcodes

Given a Galois extension we relate subfield subcodes with trace codes showing that a code is invariant under the Galois group if and only if its restriction coincides with the trace code.     

Algebraic geometric codes on anticanonical surfaces

Algebraic geometric codes (or AG codes) provide a way to correct errors that occur during the transmission of digital information. AG codes on curves have been studied extensively, but much less work has been done for AG codes on higher dimensional varieties. In particular, we seek good bounds for the minimum distance.We study AG codes on anticanonical surfaces coming from blow-ups of P² at points on a line and points on the union of two lines. We can compute the dimension of such codes exactly due to known results. For certain families of these codes, we prove an exact result on the minimum distance. For other families, we obtain lower bounds on the minimum distance.     

Galois LCD codes over finite fields

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k-Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k-Galois LCD codes is obtained, two classes of k-Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ-constacyclic codes being k-Galois LCD codes are characterized. Some classes of k-Galois LCD λ-constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ-constacyclic codes, and present a class of Hermitian LCD λ-constacyclic MDS codes.     

Minimum distance of orthogonal line-Grassmann codes in even characteristic

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for q even. The case q odd was solved in [3]. For $n\ne 3$ we also determine their second smallest distance. Furthermore, we show that for q even all minimum weight codewords are equivalent and that the symplectic line-Grassmann codes are proper subcodes of codimension 2n of the orthogonal ones.     

On secant varieties of compact Hermitian symmetric spaces

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three—with one exception, the secant variety of the 21-dimensional spinor variety in P⁶³ where we show that the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

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.     

Subspace subcodes of generalized reed-solomon codes

1. IntroductionTlle subfield subcodes of RS Cud('s a11(l tl1" tra('e-sIlurtened RS codes are introducedill [1] al1d [2], respe(!tiv'lly. II1 [3] the subsI)acc subcodes ()f RS codes over GF(2"') areintrodllced, which caIl be rcgarded as a ge11eralizatio11 of the subfield subcodes of RS codesa11d tlle trace-shorte1led RS codes, a11d tl1c fOrll1ulas t() ('oIllI)llte tl1e diInensions of thesubsPace subc(,des ()f RS c()des ()ver GF(2"' ) are given. llut what we call get if, i11stead ()fR…     

Derivation techniques on the Hermitian surface

We discuss derivation‐like techniques for transforming one locally Hermitian partial ovoid of the Hermitian surface H(3,q²) into another one. These techniques correspond to replacing a regulus by its opposite in some naturally associated projective 3‐space PG(3,q) over a square root subfield. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 478–486, 2007     

On the Twistor Spaces of Almost Hermitian Manifolds

We study Grassmannian bundles G_k(M) of analytical 2k-planes over an almost Hermitian manifold M²ⁿ, from the point of view of the generalized twistor spaces of [13], and with the method of the moving frame [9]. G₁(M⁴) is the classical twistor space. We find four distinguished almost Hermitian structures, one of them being that of [13], and discuss their integrability and Kählerianity. For n=2, we compute the corresponding Hermitian connections, and derive consequences about the corresponding first Chern classes.     

Some examples of Hermitian surfaces

The Riemannian version of the Goldberg-Sachs theorem says that a compact Einstein Hermitian surface is locally conformal Kähler. In contrast to the compact case, we show that there exists an Einstein Hermitian surface which is not locally conformal Kähler. On the other hand, it is known that on a compact Hermitian surface M ⁴, the zero scalar curvature defect implies that M ⁴ is Kähler. Contrary to the compact case, we show that there exists a non-Kähler Hermitian surface with zero scalar curvature defect.     

The relative generalized Hamming weight of linear q-ary codes and their subcodes

In this article, some properties of the relative generalized Hamming weight (RGHW) of linear codes and their subcodes are developed with techniques in finite projective geometry. The relative generalized Hamming weights of almost all 4-dimensional q-ary linear codes and their subcodes are determined.