arXiv subjects

Yu Tsunoda

Publications and source records attributed to Yu Tsunoda.

6 recordsLinked to original sources

arXiv2026

External Difference Families Arising from Two or Three Cyclotomic Classes

We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime $p=12k+1$, the family $\{C_0^6,C_2^6,C_4^6\}$ forms an EDF in $\mathbb{F}_p$ if and only if $k$ is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.

math.CO↗

arXiv2024

Weak Superimposed Codes of Improved Asymptotic Rate and Their Randomized Construction

Weak superimposed codes are combinatorial structures related closely to generalized cover-free families, superimposed codes, and disjunct matrices in that they are only required to satisfy similar but less stringent conditions. This class of codes may also be seen as a stricter variant of what are known as locally thin families in combinatorics. Originally, weak superimposed codes were introduced in the context of multimedia content protection against illegal distribution of copies under the assumption that a coalition of malicious users may employ the averaging attack with adversarial noise. As in many other kinds of codes in information theory, it is of interest and importance in the study of weak superimposed codes to find the highest achievable rate in the asymptotic regime and give an efficient construction that produces an infinite sequence of codes that achieve it. Here, we prove a tighter lower bound than the sharpest known one on the rate of optimal weak superimposed codes and give a polynomial-time randomized construction algorithm for codes that asymptotically attain our improved bound with high probability. Our probabilistic approach is versatile and applicable to many other related codes and arrays.

cs.IT↗

arXiv2024

The Asymptotics of Difference Systems of Sets for Synchronization and Phase Detection

We settle the problem of determining the asymptotic behavior of the parameters of optimal difference systems of sets, or DSSes for short, which were originally introduced for computationally efficient frame synchronization under the presence of additive noise. We prove that the lowest achievable redundancy of a DSS asymptotically attains Levenshtein's lower bound for any alphabet size and relative index, answering the question of Levenshtein posed in 1971. Our proof is probabilistic and gives a linear-time randomized algorithm for constructing asymptotically optimal DSSes with high probability for any alphabet size and information rate. This provides efficient self-synchronizing codes with strong noise resilience. We also point out an application of DSSes to phase detection.

cs.IT↗

arXiv2019

On the Maximum Number of Codewords of X-Codes of Constant Weight Three

X-codes form a special class of linear maps which were originally introduced for data compression in VLSI testing and are also known to give special parity-check matrices for linear codes suitable for error-erasure channels. In the context of circuit testing, an $(m, n, d, x)$ X-code compresses $n$-bit output data $R$ from the circuit under test into $m$ bits, while allowing for detecting the existence of an up to $d$-bit-wise anomaly in $R$ even if up to $x$ bits of the original uncompressed $R$ are unknowable to the tester. Using probabilistic combinatorics, we give a nontrivial lower bound for any $d \geq 2$ on the maximum number $n$ of codewords such that an $(m, n, d, 2)$ X-code of constant weight $3$ exists. This is the first result that shows the existence of an infinite sequence of X-codes whose compaction ratio tends to infinity for any fixed $d$ under severe weight restrictions. We also give a deterministic polynomial-time algorithm that produces X-codes that achieve our bound.

cs.IT↗

arXiv2018

Bounds on Separating Redundancy of Linear Codes and Rates of X-Codes

An error-erasure channel is a simple noise model that introduces both errors and erasures. While the two types of errors can be corrected simultaneously with error-correcting codes, it is also known that any linear code allows for first correcting errors and then erasures in two-step decoding. In particular, a carefully designed parity-check matrix not only allows for separating erasures from errors but also makes it possible to efficiently correct erasures. The separating redundancy of a linear code is the number of parity-check equations in a smallest parity-check matrix that has the required property for this error-erasure separation. In a sense, it is a parameter of a linear code that represents the minimum overhead for efficiently separating erasures from errors. While several bounds on separating redundancy are known, there still remains a wide gap between upper and lower bounds except for a few limited cases. In this paper, using probabilistic combinatorics and design theory, we improve both upper and lower bounds on separating redundancy. We also show a relation between parity-check matrices for error-erasure separation and special matrices, called X-codes, for data compaction circuits in VLSI testing. This leads to an exponentially improved bound on the size of an optimal X-code.

cs.IT↗

arXiv2016

Probabilistic bounds on the trapping redundancy of linear codes

The trapping redundancy of a linear code is the number of rows of a smallest parity-check matrix such that no submatrix forms an $(a,b)$-trapping set. This concept was first introduced in the context of low-density parity-check (LDPC) codes in an attempt to estimate the number of redundant rows in a parity-check matrix suitable for iterative decoding. Essentially the same concepts appear in other contexts as well such as robust syndrome extraction for quantum error correction. Among the known upper bounds on the trapping redundancy, the strongest one was proposed by employing a powerful tool in probabilistic combinatorics, called the Lovász Local Lemma. Unfortunately, the proposed proof invoked this tool in a situation where an assumption made in the lemma does not necessarily hold. Hence, although we do not doubt that nonetheless the proposed bound actually holds, for it to be a mathematical theorem, a more rigorous proof is desired. Another disadvantage of the proposed bound is that it is only applicable to $(a,b)$-trapping sets with rather small $a$. Here, we give a more general and sharper upper bound on trapping redundancy by making mathematically more rigorous use of probabilistic combinatorics without relying on the lemma. Our bound is applicable to all potentially avoidable $(a,b)$-trapping sets with $a$ smaller than the minimum distance of a given linear code, while being generally much sharper than the bound through the Lovász Local Lemma. In fact, our upper bound is sharp enough to exactly determine the trapping redundancy for many cases, thereby providing precise knowledge in the form of a more general bound with mathematical rigor.

cs.IT↗