arXiv subjects

Sihuang Hu

Publications and source records attributed to Sihuang Hu.

32 records · Page 2Linked to original sources

arXiv2022

Extended Cyclic Codes Sandwiched Between Reed-Muller Codes

The famous Barnes-Wall lattices can be obtained by applying Construction D to a chain of Reed-Muller codes. By applying Construction ${D}^{(cyc)}$ to a chain of extended cyclic codes sandwiched between Reed-Muller codes, Hu and Nebe (J. London Math. Soc. (2) 101 (2020) 1068-1089) constructed new series of universally strongly perfect lattices sandwiched between Barnes-Wall lattices. In this paper, we first extend their construction to generalized Reed-Muller codes, and then explicitly determine the minimum vectors of those new sandwiched Reed-Muller codes for some special cases.

cs.IT↗

arXiv2021

A Dynamic Programming Method to Construct Polar Codes with Improved Performance

In the standard polar code construction, the message vector $(U_0,U_1,\dots,U_{n-1})$ is divided into information bits and frozen bits according to the reliability of each $U_i$ given $(U_0,U_1,\dots,U_{i-1})$ and all the channel outputs. While this reliability function is the most suitable measure to choose information bits under the Successive Cancellation (SC) decoder, there is a mismatch between this reliability function and the Successive Cancellation List (SCL) decoder because the SCL decoder also makes use of the information from the future frozen bits. We propose a Dynamic Programming (DP) construction of polar codes to resolve this mismatch. Our DP construction chooses different sets of information bits for different list sizes in order to optimize the performance of the constructed code under the SCL decoder. Simulation results show that our DP-polar codes consistently demonstrate $0.3$--$1$dB improvement over the standard polar codes under the SCL decoder with list size $32$ for various choices of code lengths and code rates.

cs.IT↗

arXiv2020

Relative projective group codes over chain rings

A structure theorem of the group codes which are relative projective for the subgroup $\lbrace 1 \rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $\ell$ in the group algebra $\mathbb{F}G$, where $\mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight.

cs.IT↗

arXiv2019

Low dimensional strongly perfect lattices IV: The dual strongly perfect lattices of dimension 16

We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $Λ_{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its dual, and one pair of new lattices $Γ_{16}$ and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices.

math.NT↗

arXiv2018

On the VC-Dimension of Binary Codes

We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $δn$. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening approach combining Sauer-Shelah lemma and the linear programming bound. Two lower bounds are given using Gilbert-Varshamov type arguments over constant-weight and Markov-type sets.

cs.IT↗

arXiv2018

Strongly perfect lattices sandwiched between Barnes-Wall lattices

New series of $2^{2m}$-dimensional universally strongly perfect lattices $Λ_I $ and $Γ_J $ are constructed with $$2BW_{2m} ^{\#} \subseteq Γ_J \subseteq BW_{2m} \subseteq Λ_I \subseteq BW _{2m}^{\#} .$$ The lattices are found by restricting the spin representations of the automorphism group of the Barnes-Wall lattice to its subgroup ${\mathcal U}_m:={\mathcal C}_m (4^H_{\bf 1}) $. The group ${\mathcal U}_m$ is the Clifford-Weil group associated to the Hermitian self-dual codes over ${\bf F} _4$ containing ${\bf 1}$, so the ring of polynomial invariants of ${\mathcal U}_m$ is spanned by the genus-$m$ complete weight enumerators of such codes. This allows us to show that all the ${\mathcal U}_m$ invariant lattices are universally strongly perfect. We introduce a new construction, $D^{(cyc)}$ for chains of (extended) cyclic codes to obtain (bounds on) the minimum of the new lattices.

math.NT↗

arXiv2018

A Bound on the Shannon Capacity via a Linear Programming Variation

We prove an upper bound on the Shannon capacity of a graph via a linear programming variation. We show that our bound can outperform both the Lovász theta number and the Haemers minimum rank bound. As a by-product, we also obtain a new upper bound on the broadcast rate of Index Coding.

cs.IT↗

arXiv2018

Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes

Locally recoverable (LRC) codes have recently been a focus point of research in coding theory due to their theoretical appeal and applications in distributed storage systems. In an LRC code, any erased symbol of a codeword can be recovered by accessing only a small number of other symbols. For LRC codes over a small alphabet (such as binary), the optimal rate-distance trade-off is unknown. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The resulting LP bound gives better estimates in examples than the other upper bounds known in the literature. Further, we provide the tightest known upper bound on the rate of linear LRC codes with a given relative distance, an improvement over the previous best known bounds.

cs.IT↗

arXiv2017

Quickest Sequence Phase Detection

A phase detection sequence is a length-$n$ cyclic sequence, such that the location of any length-$k$ contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible $k$ in the limit of $n\to\infty$, and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length-$k$ contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed.

cs.IT↗

arXiv2017

The $ρ$-Capacity of a Graph

Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed $ρ$-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the $ρ$-capacity of arbitrary graphs, and provide a Lovász-type upper bound for regular graphs. We study the behavior of the $ρ$-capacity under two graph operations: the strong product and the disjoint union. Finally, we investigate the connection between the structure of a graph and its $ρ$-capacity.

cs.IT↗

arXiv2014

New pseudo-planar binomials in characteristic two and related schemes

Planar functions in odd characteristic were introduced by Dembowski and Ostrom in order to construct finite projective planes in 1968. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and signal sets. Recently, a new notion of pseudo-planar functions in even characteristic was proposed by Zhou. These new pseudo-planar functions, as an analogue of planar functions in odd characteristic, also bring about finite projective planes. There are three known infinite families of pseudo-planar monomial functions constructed by Schmidt and Zhou, and Scherr and Zieve. In this paper, three new classes of pseudo-planar binomials are provided. Moreover, we find that each pseudo-planar function gives an association scheme which is defined on a Galois ring.

math.CO↗

arXiv2013

Difference Sets with Few Character Values

The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd$(v,n)>1$. It is remarkable that all the known difference sets with gcd$(v,n)>1$ have the so-called character divisibility property. In 1997, Jungnickel and Schmidt posed the problem of constructing difference sets with gcd$(v,n)>1$ that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.

math.CO↗

arXiv2012

The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs

The determination of weight distribution of cyclic codes involves evaluation of Gauss sums and exponential sums. Despite of some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectrums of Hermitian forms graphs.

cs.IT↗

arXiv2012

Association schemes related to Delsarte-Goethals codes

In this paper, we construct an infinite series of 9-class association schemes from a refinement of the partition of Delsarte-Goethals codes by their Lee weights. The explicit expressions of the dual schemes are determined through direct manipulations of complicated exponential sums. As a byproduct, the other three infinite families of association schemes are also obtained as fusion schemes and quotient schemes.

math.CO↗