Abstracts

• Jun Zhang

A class of twisted generalized Reed-Solomon codes

Twisted generalized Reed-Solomon (TGRS) codes were proposed by Beelen, Puchinger, Nielsen in 2017 and have been attracted a lot of attention. In this talk, we consider a class of TGRS codes. Their dual codes and minimum distances are determined. Self-dual TGRS codes are presented. This work is jointed with Chunming Tang and Zhengchun Zhou.

• Jon-Lark Kim

Codes over a non-commutative non-unital ring E with four elements

The ring E is one of the 11 rings with four elements. It is not commutative and does not have a unity. Nevertheless, we can study linear codes over E. In this talk, we describe the current status of quasi-self-dual codes and LCD codes over E.

• Yajing Zhou

Symmetrical Z-complementary code sets for optimal training in generalized spatial modulation

In this talk, we consider the optimal training design for broadband generalized spatial modulation systems over frequency-selective channels. We introduce a novel class of code sets, called ``symmetrical Z-complementary code sets'', whose aperiodic auto- and cross- correlation sums exhibit zero-correlation zones at both the front-end and tail-end of the entire correlation window. Two constructions of (optimal) symmetrical Z-complementary code sets based on generalized Boolean functions are presented. Numerical evaluations indicate that the proposed sequences lead to minimum channel estimation mean-square error and significantly outperform other classes of sequences. This work is jointed with Zhengchun Zhou, Zilong Liu, Yang Yang, Ping Yang and Pingzhi Fan.

• Chengju Li

The dual codes of several classes of BCH codes

A BCH code of length $n$ over $\mathbb{F}_q$ is always relative to an $n$-th primitive root of unity $\beta$ in an extension field of $\mathbb{F}_q$, and is called a dually-BCH code if its dual is also a BCH code relative to the same $\beta$. In this talk, we will give sufficient and necessary conditions in terms of the designed distances $\delta$ to ensure that BCH codes are dually-BCH codes for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes. In addition, the parameters of these BCH codes and their dual codes are investigated. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated.

• Yeongwook Kwon

MacWilliams identities for some weight enumerators in higher genus of linear codes over a Frobenius ring of order 16

In this talk, we establish a connection between Jacobi forms and codes over a Frobenius ring of order 16. One of the main features of this work is that the related Jacobi forms are over totally real fields. Moreover, we introduce MacWilliams identities for some weight enumerators in higher genera. Finally, we give invariants via a self-dual code of even length. This is joint work with Boran Kim, Chang Heon Kim and Soonhak Kwon.

• Deng Tang

Constructions of binary locally repairable codes with locality two and multiple repair alternatives via autocorrelation spectra of Boolean functions

Distributed storage systems (in brief, DSSs) store data on several distributed nodes and are widely used in file system storage, large database storage, backup file, and cloud storage, etc. DSSs provide reliable access to data through redundancy spread over individually unreliable nodes, where the replication scheme and coding mechanism are two widespread techniques for ensuring reliability. In 2012, Gopalan et al. proposed locally repairable codes (LRCs for short) to minimize the number of nodes to be downloaded in repairing any node. A code over a finite alphabet is called LRC (with locality $r$) if every symbol in the encoding is a function of a small number (at most $r$) of other symbols of the codeword. In 2013 Pamies-Juarez et al. introduced LRCs with multiple repair alternatives, which allows repairing any node with different disjoint nodes. LRCs with multiple repair alternatives can increase the probability of being able to perform efficient repairs when there are multiple unavailable nodes (these nodes are failed or temporarily unavailable). This talk proposes three large families of LRCs with multiple repair alternatives from Boolean functions. Each repair set has at most $r=2$ symbols, which correspond to an interesting case in practice. We shall explore Boolean functions selected from some well-known constructions. Moreover, we show that the number of the disjoint repair sets (denoted by $t$) of our LRCs can be determined entirely by the autocorrelation spectrum of the corresponding Boolean function. This achievement is obtained thanks to the relationship between the autocorrelation spectrum of the corresponding Boolean function and the number of disjoint repairs sets that we establish. Our results give rise LRCs with suitable parameters from special Boolean functions (such as bent functions) based on a construction method introduced by Ding in 2015 for designing linear codes based on the so-called ``defining set" (involving mainly Boolean functions). The approach presented in this talk introduces an interesting connection between LRCs (with multiple repair alternatives) and (the autocorrelation spectrum of) Boolean functions.

• Zhi Jing

Some new sequential-recovery LRCs based on good polynomials

We proposed a new construction of sequential-recovery LRCs of length n with even locality r for two erasures, based on some "good" polynomial. We also derive an explicit form of the upper bound on the minimum distance of these codes with some additional constraints. The minimum distance of the proposed sequential-recovery LRCs for r=2 achieves this explicit bound when n=2k and is one less than the bound when n>2k.

• Xiande Zhang

Results on ternary constant weight codes under l1-metric

In this talk, we discuss our recent progress on the existence of optimal ternary constant weight codes in l1-metric. The motivation of studying constant weight codes in l1-metric is from data storage in live DNA.

• Jong Yoon Hyun

Ramanujan graphs and expander families constructed from p-ary bent functions

We present a method for constructing an infinite family of non-bipartite Ramanujan graphs. We mainly employ p-ary bent functions of ( p − 1)-form for this construction, where p is a prime number. Our result leads to construction of infinite families of expander graphs; this is due to the fact that Ramanujan graphs play as base expanders for constructing further expanders. For our construction we directly compute the eigenvalues of the Ramanujan graphs arsing from p-ary bent functions. Furthermore, we establish a criterion on the regularity of p-ary bent functions in m variables of ( p − 1)-form when m is even. Finally, using weakly regular p-ary bent functions of (\ell-1)-form, we find (amorphic) association schemes in a constructive way; this resolves the open case that \ell= p − 1for p > 2 for finding (amorphic) association schemes. This is joint work with Jungyun Lee (Kangwon National University), and Yoonjin Lee (Ewha Womans University).

• Nian Li

A subfield-based construction of optimal linear codes over finite fields

Linear codes have a wide range of applications in the data storage systems, communication systems and consumer electronics products. In this talk, we present four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distanceoptimal linear codes. In addition, by a more in-depth discussion on some special cases of these four families of linear codes, we obtain several classes of (distance-)optimal linear codes with few weights and completely determine their weight distributions. We also show that most of our linear codes are self-orthogonal or minimal which are useful in applications.

• Sunghyu Han

On the construction of MDS self-dual codes over Galois rings

We study the MDS self-dual codes over finite chain rings. We state the projection and lifting of codes over the finite chain rings with respect to the MDS self-dual codes, and then we apply the results to the MDS self-dual codes over Galois rings. In addition, we give some constructions of MDS self-dual codes over Galois rings with even characteristic.

• Yiwei Zhang

t-deletion-1-insertion-burst correcting codes

The history of codes against deletion and insertion errors dates back to the celebrated Varshamov-Tenengolts code. In recent years there are many new progress on codes against deletions and insertions, mainly due to their applications in DNA storage. In this talk, we introduce a code against a specific type of error, known as $t$-del-$1$-ins-burst,which deletes a consecutive substring of length $t$ and then inserts an arbitrary symbol at the same coordinate.We introduce our recent results on such codes, including a sphere-packing type upper bound and some explicit constructions. This is a joint work with my student Mr. Ziyang Lu.

• Junyong Park

Steganography from perfect codes on Cayley graphs over Gaussian integers, Eisenstein-Jacobi integers and Lipschitz integers

Steganography is the science of communicating a secret message by hiding it in a cover object. It may be an interesting research problem to construct some steganographic schemes from mathematically structured graphs. In this talk, we study to construct steganographic schemes explicitly from r-perfect codes on Cayley graphs over Gaussian integers, Eisenstein-Jacobi integers, and Lipschitz integers, respectively. Then we further compute various parameters on the suggested steganographic schemes.

• Yaqian Zhang

An improved cooperative repair scheme for Reed-Solomon codes

Dau et al. extend Guruswami and Wootters’ scheme (STOC’2016) to cooperatively repair two or three erasures in Reed-Solomon (RS) codes. However, their schemes restrict to either the case that the characteristic of the finite field F divides the extension degree [F : B] or some special failure patterns, where F is the base field of the RS code and B is the subfield of the repair symbols. In this report, we give an improved cooperative repair scheme for RS codes that remove all these restrictions. Specifically, we obtain a one-round cooperative repair scheme for any two erasures and a three-round scheme for any three erasures. When restricted to a wide class of failure patterns, we also achieve a one-round cooperative repair scheme for three erasures. The repair bandwidth of our schemes remains the same with that of Dau et al.’s schemes.

• Young Gun Roe

Construction of quasi self-dual codes over a commutative non-unital ring of order 4

Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6, and suggested two building-up methods for constructing QSD codes. In this talk, we show constructions of more QSD codes, Type IV codes and quasi Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods.

• Minjia Shi

An improved method for constructing linear codes with small hulls

we give an improved method for constructing linear codes with small hulls. As a result, we obtain many optimal Euclidean LCD codes and Hermitian LCD codes, which improve the previously known lower bound on the largest minimum distance. We also obtain many optimal codes with one-dimension hull. Furthermore, we give three tables about formally self-dual LCD codes.

• Eun Ju Cheon

Some constructions of minimal linear codes

In 2021, for a prime p, Hyun et al defined p-ary linear codes by using multi-variable functions, and constructed three classes of minimal p-ary linear codes. We extend their results to q-ary case. We constructed some minimal q-ary linear codes as a generalization of their minimal p-ary linear codes.

• Qi Wang

Non-overlapping codes

Non-overlapping codes have been studied for decades, and have recently found important applications in DNA storage systems. In this talk, I will first survey important results on non-overlapping codes, and then talk about some recent new results.

• Chunming Tang

On infinite families of narrow-sense antiprimitive BCH codes admitting 3-transitive automorphism

The Bose-Chaudhuri-Hocquenghem(BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantumin formation processing.They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field GF(q) with length q+1, which are closely related to the action of the projective general linear group of degree two on the projective line.Despite its interest, not much is known about this class of BCH codes. In this talk we aim to study some of the codes within this class and specifically narrow-senseanti primitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from there centliterature. Notably, the dimension, the minimum distance of some q-ary BCH codes with length q+1, and their duals are determined. The dual codes of the narrow-sense antiprimitive BCH codes derived include almost MDS codes. Furthermore, the classification of PGL (2, pm)-invariant codes over GF(ph) is completed. As an application of this result, the p-ranks of all incidence structures invariant under the projective general linear group PGL (2, pm) are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, acoding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this talk are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

• Whan-Hyuk Choi

Self-orthogonality matrix and Reed-Muller codes

Kim et al. (2021) gave a method to embed a given binary [𝑛, 𝑘] code C (𝑘 = 3, 4) into a self-orthogonal code of the shortest length which has the same dimension 𝑘 and minimum distance 𝑑. We extends this result for 𝑘 = 5 and 6 by proposing a new method related to a special matrix, called the self-orthogonality matrix 𝑆𝑂𝑘, obtained by shortnening a Reed-Muller code R (2, 𝑘). Furthermore, we disprove partially the conjecture (Kim et al. (2021)) by showing that if 31 ≤ 𝑛 ≤ 256 and 𝑛 ≡ 14, 22, 29 (mod 31), then there exist optimal [𝑛, 5] codes which are self-orthogonal. We also construct optimal self-orthogonal [𝑛, 6] codes when 41 ≤ 𝑛 ≤ 256 satisfies 𝑛 ≠ 46, 54, 61 and 𝑛. 7, 14, 22, 29, 38, 45, 53, 60 (mod 63).

• Zhengchun Zhou

Low ambiguity zone: Theoretical bounds and Doppler-resilient sequence design in integrated sensing and communication systems

In radar sensing and communications, designing Doppler resilient sequences (DRSs) with low ambiguity function for delay over the entire signal duration and Doppler shift over the entire signal bandwidth is an extremely difficult task. However, in practice, the Doppler frequency range is normally much smaller than the bandwidth of the transmitted signal, and it is relatively easy to attain quasi-synchronization for delays far less than the entire signal duration. In this talk, we shall introduce a new concept called low ambiguity zone (LAZ) which is a small area of the corresponding ambiguity function of interest defined by the certain Doppler frequency and delay. Such an LAZ will reduce to a zero ambiguity zone (ZAZ) if the maximum ambiguity values of interest are zero. We will introduce a set of theoretical bounds on periodic LAZ/ZAZ of unimodular DRSs with and without spectral constraints, which include the existing bounds on periodic global ambiguity function as special cases. These bounds may be used as theoretical design guidelines to measure the optimality of sequences against Doppler effect. We will also present four optimal constructions of DRSs with respect to the derived ambiguity lower bounds based on some algebraic tools such as characters over finite field and cyclic difference sets.

• Jieun Kwon

Desings and linear codes associated with r-plateaued functions

Walsh transformation is a very important concept to define r-plateaued functions which can be considered as a concept of generalization of bent functions. Also, the block designs induced by bent functions were studied for decades. In this talk, we generalize some properties of designs associated with bent functions to designs associated with r-plateaued functions. And we suggest the way of calculate the weight distribution of some codes through the value of Walsh transformation.