We study Schubert subspace codes, which are constant-dimension subspace codes with prescribed intersection conditions with a fixed subspace. Our goal is to construct codes of maximum possible size in the extremal distance cases where a natural counting upper bound applies. We give two families of constructions. The first one uses a direct-sum decomposition of the ambient space, together with partial spreads and colorings of powers of $q$-Johnson graphs. For this construction, we also prove necessary conditions, which show how chromatic and clique obstructions arise. The second family is obtained by field reduction from evasive and scattered subspaces over extension fields. This gives codes whose size can be computed exactly in the scattered case and recovers the only previously known construction as a special case.
We provide a positive answer to a conjecture proposed by Toh\v{a}neanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classical B\'ezout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.
Minimum distance bounds play a central role in the analysis of algebraic codes. For cyclic and constacyclic codes, several bounds based on their zero set have been developed. However, analogous results for quasi-twisted (QT) codes are comparatively limited. In this paper, we further investigate the spectral theory of QT codes and derive a general spectral bound on their minimum distance. Our bound unifies and generalizes previously known spectral bounds for quasi-cyclic (QC) and QT codes, and contains them as special cases. We present a new proof technique and show that the bound can be formulated with respect to an arbitrary subset of eigenvalues, thereby extending its applicability to the largest possible setting. Numerical examples and simulations demonstrate that the proposed bound often improves upon the Jensen bound and earlier spectral bounds.
Unknown authors· Turkish Journal of Mathemati...· 0 citations
It is demonstrated that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.
In 2012, Kaufman and Lubotzky constructed the first family of symmetric LDPC good codes. Their construction used Cayley codes, as originally defined by Kaufman and Wigderson (2016). In this paper we present two generalisations to the Cayley code construction, which we call graph codes and digraph codes. We investigate both the algebraic, and combinatorial properties of these constructions and show that they possess the same desirable attributes as Cayley codes, but with added freedom. We analyse the relationship between the expansion properties of the ingredient (di)graphs and the parameters of the constructed codes; our analysis offers an improvement to the results of Kaufman and Lubotzky. As an application, we construct an infinite family of good digraph codes, and we propose a series of open problems.
Coen del Valle, C. Praeger· arXiv.org· 0 citations
We study upper bounds on the length of $\mathbb F_q$-linear QMDS codes in the folded Hamming distance relative to their other parameters, especially the field size $q$. Via a correspondence between such codes and families of subspaces, we relate the length problem to that of upper bounding $1$-subspace packings with respect to the other parameters, especially the field size. Our main result is a reduction from these families to partial spreads, which allows us to import sharp bounds from finite geometry, including results of Drake-Freeman, N\u{a}stase-Sissokho, and Honold-Kiermaier-Kurz. As a consequence, we recover the Griesmer-type upper bound on the length of QMDS codes by Ball et al. and obtain tighter upper bounds in several parameter regimes.
In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.
Antonio Jesús Lorite-López, Daniel Camaz'on-Portela, J. A. López-Ramos· 0 citations
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