arXiv subjects
Publications and source records attributed to Keita Ishizuka.
We extend the complete classifications of linear complementary dual (LCD) codes from length 15 to length 19 in the binary case, from length 10 to length 15 in the ternary case and from length 12 to length 13 in the quaternary Hermitian case. The classification rests on a reduction of the monomial equivalence of LCD codes to graph isomorphism on the coordinates, combined with canonical augmentation.
We establish a connection between linear complementary dual (LCD) codes and caps in projective space. Using this framework and the structure theory of maximal binary caps, we derive nonexistence theorems for binary LCD codes with minimum distance at least $4$, providing computation-free proofs that were previously obtained only through exhaustive search. As an application, we completely determine the optimal minimum distances for codimensions $7$ and $8$ for the first time. The framework applies to ternary LCD codes and quaternary Hermitian LCD codes, where it reduces the problem to the structure of large caps. Using known results on large caps, we determine the optimal minimum distances of ternary LCD codes of codimension $6$ for all lengths, and we explain why the phenomena behind the binary nonexistence theorems, such as the dependence on the parity of the length, have no counterpart for these codes.
The hull of a linear code is the intersection of the code with its dual and depends on the choice of inner product. We study how the numbers of linear codes with prescribed Euclidean, Hermitian, and symplectic hull dimensions vary with the hull dimension. Building on closed enumeration formulae, we derive exact ratio decompositions for the Hermitian and symplectic cases, determine the parameter ranges in which monotonicity holds, and identify the exceptional families. We also compare the three inner products and complement the theoretical analysis with computational classification results for quaternary codes.
The Permutation Equivalence Problem (PEP) for linear complementary dual (LCD) codes reduces to Graph Isomorphism (GI) through the orthogonal projector onto the code, but this requires a trivial hull. We ask whether the reduction extends when the standard inner product is replaced by a nondegenerate symmetric bilinear form \(M\). We make precise what it means for \(M\) to yield a valid reduction to GI and prove that, for length \(n \geq 3\) and every field size, this holds if and only if \(M\) is permutation-invariant. The reduction then reaches, beyond LCD codes, a class of codes with one-dimensional hull: exactly those whose hull is spanned by a vector with nonzero coordinate sum. We count these codes exactly; for odd \(q\), asymptotically the method covers a fraction \(\prod_{i \geq 2}(1+q^{-i})^{-1}\) of all codes, \(1 + q^{-1}\) times the fraction of LCD codes. A polynomial-time reduction algorithm is given.
We prove that asymptotically almost all vectorial functions over finite fields have trivial extended-affine stabilizers. As a consequence, the number of EA-equivalence classes is asymptotically equal to the naive estimate, namely the total number of functions divided by the size of the EA-group, with vanishing relative error. Furthermore, we derive upper bounds on collision probabilities for both extended-affine and CCZ equivalences. For EA-equivalence, we leverage the trivial-stabilizer result to establish a matching lower bound, yielding a tight asymptotic formula that shows two independently sampled functions are EA-equivalent with super-exponentially small probability. The results validate random sampling strategies for cryptographic primitive design and show that functions with nontrivial EA-stabilizers form an exponentially rare subset.
We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the intersection array parameters. This necessary and sufficient criterion strengthens the previously known sufficient conditions and unifies the cases of complete, Hamming, Johnson, and Grassmann graphs. As further applications, we prove that non-isomorphic conference graphs with $q \equiv 1 \pmod 8$ yield inequivalent codes and we classify all simple graphs with idempotent adjacency matrices on at most $13$ vertices via mass formulas for binary LCD codes.
Recently, the notions of self-orthogonal subspace codes and LCD subspace codes were introduced, and LCD subspace codes obtained from mutually unbiased weighing matrices were studied. In this paper, we provide a method of constructing self-orthogonal and LCD subspace codes from a set of matrices under certain conditions. In particular, we give constructions of self-orthogonal and LCD subspace codes from mutually quasi-unbiased weighing matrices, linked systems of symmetric designs, and linked systems of symmetric group divisible designs, Deza graphs and their equitable partitions.
In this note, we give basic properties of ternary four-negacirculant self-dual codes. By exhaustive computer search based on the properties, we complete a classification of ternary extremal four-negacirculant self-dual codes of lengths 40, 44, 48, 52 and 60.
In this note, we study the existence of Hadamard matrices of order $36$ formed by codewords of weight $36$ in some ternary near-extremal self-dual codes of length $36$.
We introduce a general construction of many Hermitian LCD $[n, k]$ codes from a given Hermitian LCD $[n, k]$ code. Furthermore, we present some results on punctured codes and shortened codes of quaternary Hermitian LCD codes. As an application, we improve some of the previously known lower bounds on the largest minimum weights of quaternary Hermitian LCD codes of length $12 \le n \le 30$.
From a given $[n, k]$ code $C$, we give a method for constructing many $[n, k]$ codes $C'$ such that the hull dimensions of $C$ and $C'$ are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even $[56, 28, 12]$ codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length $n=26,28 \le n \le 40$.
For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance greater than or equal to $2$, then there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance $1$. As a consequence, we generalize a result by Araya, Harada and Saito on the nonexistence of some quaternary Hermitian linear complementary dual codes.
Hermitian linear complementary dual codes are linear codes whose intersection with their Hermitian dual code is trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$.
This paper briefly reports our ongoing attempt at the development of a multi-platform browser-based speech recording system. We designed the system toward a service of providing open service of building large-scale speech corpora at a low-cost for any researchers and developers related to speech processing. The recent increase in the use of crowdsourcing services, e.g., Amazon Mechanical Turk, enable us to reduce the cost of collecting speakers in the web, and there have been many attempts to develop the automated speech collecting platforms or application that is designed for the use the crowdsourcing. However, one of the major problems in the previous studies and developments for the attempts is that most of the systems are not a form of common service of speech recording and corpus building, and each corpus builder is necessary to develop the system in their own environment including a web server. For this problem, we develope a new platform where both the corpus builders and recording participants can commonly use a single system and service by creating their user accounts. A brief introduction of the system is given in this paper as the start of this challenge.