Thomas Honold

ZJU-UIUC Institute, Zhejiang University

We present various new constructions and bounds for arcs in projective Hjelmslev planes over finite chain rings of nilpotency index 2. For the chain rings of cardinality at most 25 we give updated tables with the best known upper and lower bounds for the maximum size of such arcs.

The problem of determining the largest possible number of distinct Hamming weights in several classes of codes over finite fields was studied recently in several papers (Shi et al. in Des Codes Cryptogr 87(1):87–95, 2019, in IEEE Trans Inf Theory 66(11):6855–6862, 2020; Chen et al. in IEEE Trans Inf Theory 69(2):995–1004, 2022). A further problem i...

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a rel...

A projective linear code over $\mathbb{F}_q$ is called $\Delta$-divisible if all weights of its codewords are divisible by $\Delta$. Especially, $q^r$-divisible projective linear codes, where $r$ is some integer, arise in many applications of collections of subspaces in $\mathbb{F}_q^v$. One example are upper bounds on the cardinality of partial sp...

It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 16, 30, 31, 32, 45–51, and ≥60.

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.

The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In Finite Geometry terms the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a...

It is shown that there does not exist a binary projective triply-even code of length $59$. This settles the last open length for projective triply-even binary codes. Therefore, projective triply-even binary codes exist precisely for lengths $15$, $16$, $30$, $31$, $32$, $45$--$51$, and $\ge 60$.

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is...

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake & Freeman on partial spreads date back to 1...

We give a sufficient condition for a bi-invariant weight on a Frobenius bimodule to satisfy the extension property. This condition applies to bi-invariant weights on a finite Frobenius ring as a special case. The complex-valued functions on a Frobenius bimodule are viewed as a module over the semigroup ring of the multiplicative semigroup of the co...

The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In finite geometry terms the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a...

For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $\mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The motivation for studying this problem comes from the theory of partial spreads, or subspace codes with the highest possi...

Constant dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in finite geometry for several decades. It is no surprise that the sharpest bounds on the maximal code sizes are typically known for this subclass. The seminal works of Andr\'e, Segre, Beutelspacher, and Drake \& Freeman date back...

A general approach is established for deriving one-shot performance bounds for information-theoretic problems on general alphabets beyond countable alphabets. It is mainly based on the quantization idea and a novel form of “likelihood ratio”. As an example, one-shot lower and upper bounds for random number generation from correlated sources on gene...

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and...

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and...

A set Fq of 3-dimensional subspaces of F7q , the 7-dimensional vector space over the finite field Fq, is said to form a q-analogue of the Fano plane if every 2-dimensional subspace of F7q is contained in precisely one member of Fq. The existence problem for such q-analogues remains unsolved for every single value of q. Here we report on an attempt...

A general approach, as an extension of the information-spectrum approach, is established for deriving one-shot performance bounds for information-theoretic problems whose alphabets are Polish. It is mainly based on the quantization idea and a novel form of "likelihood ratio". As an example, one-shot lower and upper bounds for random number generati...

As shown in [28], one of the five isomorphism types of optimal binary subspace codes of size 77 for packet length v=6, constant dimension k=3 and minimum subspace distance d=4 can be constructed by first expurgating and then augmenting the corresponding lifted Gabidulin code in a fairly simple way. The method was refined in [32,26] to yield an esse...

Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. The resulting so-called Main Problem of Subspace Coding is to determine the maximum size Aq(υ, d) of a code in PG(υ –1; Fq) with minimum subspace distance d. Here we completely resolve this problem for d ≥ v –...

A set $\mathcal{F}_q$ of $3$-dimensional subspaces of $\mathbb{F}_q^7$, the $7$-dimensional vector space over the finite field $\mathbb{F}_q$, is said to form a $q$-analogue of the Fano plane if every $2$-dimensional subspace of $\mathbb{F}_q^7$ is contained in precisely one member of $\mathcal{F}_q$. The existence problem for such $q$-analogues re...

Maximal arcs in small projective Hjelmslev geometries are classified up to isomorphism, and the parameters of the associated codes are determined.

Subspace codes form the appropriate mathematical setting for investigating the Koetter-Kschischang model of fault-tolerant network coding. The Main Problem of Subspace Coding asks for the determination of a subspace code of maximum size (proportional to the transmission rate) if the remaining parameters are kept fixed. We describe a new approach to...

The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2]. Secondly, these codes are used to define a dual two-weight code and strongly regular graph similar to the classi...

The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2]. Secondly, these codes are used to define a dual two-weight code and strongly regular graph similar to the classi...

It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$ mutually intersecting in at most a point is $77$. Optimal binary $(v,M,d;k)=(6,77,4;3)$ subspace codes are cla...

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming...

We prove that (q 2, 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q 2 and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q 2 + 2 on the maximum size of a 2...

Let R be a chain ring with four elements. In this paper, we present two new constructions of R-linear codes that contain a subcode associated with a simplex code over the ring R. The simplex codes are defined as the codes generated by a matrix having as columns the homogeneous coordinates of all points in some projective Hjelmslev geometry PHG(R k...

The random matrix uniformly distributed over the set of all m-by-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is...

Cyclic Redundancy Check (CRC) is a well-established coding method for error detection and is part of several industrial standards for data transmission and storage. If a second CRC is used on top of an existing standard CRC to provide additional reliability, we call this an embedded CRC. The performance of such CRC combinations has been considered...

We complete the determination of the maximum sizes of (k, n)-arcs, n ≤ 12, in the projective Hjelmslev planes over the two (proper) chain rings ℤ9 = ℤ/9ℤ and $\mathbb{S}_3 = \mathbb{F}_3 {{[X]} \mathord{\left/ {\vphantom {{[X]} {(X^2 )}}} \right. \kern-0em} {(X^2 )}}$ of order 9 by resolving the hitherto open cases n = 6 and n = 7. Parts of our p...

The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the aver...

Linear operator broadcast channel (LOBC) models the scenario of multi-rate packet broadcasting over a network, when random network coding is applied. This paper presents the framework of algebraic coding for LOBCs and provides a Hamming-like upper bound on (multishot) subspace codes for LOBCs.

In this paper, we present a duality construction for multiarcs in projective Hjelmslev geometries over chain rings of nilpotency index 2. We compute the parameters of the resulting arcs and discuss some examples.

Recent research indicates that packet transmission employing random linear network coding can be regarded as transmitting subspaces over a linear operator channel (LOC). In this paper we propose the framework of linear operator broadcast channels (LOBCs) to model packet broadcasting over LOCs, and we do initial work on the capacity region of consta...

A set S of points in a finite incidence structure is said to be a two-intersection set if there are integers a < b such that S meets every block in either a or b points (and both a, b actually occur as intersection numbers). For point-hyperplane designs of the classical geometries PG(k, q) such sets have been studied extensively and related to othe...

Linear encoders with good joint spectra are good candidates for lossless joint source-channel coding (JSCC), where the joint spectrum is a variant of the input-output complete weight distribution and is considered good if it is close to the average joint spectrum of all linear encoders (of the same coding rate). However, in spite of their existence...

This paper presents a cross-layer iterative decoder for irregular low-density parity-check (LDPC) codes which uses cyclic redundancy check (CRC) codes. The key idea of the decoder is to use correctly decoded frames as an aid for correcting the remaining erroneous frames. To accomplish this, the decoder exchanges the relevant information between lay...

In this paper, we present various results on arcs in projective three- dimensional Hjelmslev spaces over nite chain rings of nilpotency index 2. A table is given containing exact values and bounds for projective arcs in the geometries over the two chain rings with four elements.

Linear codes with good joint spectra are good candidates for lossless joint source-channel coding (JSCC). However, in spite of their existence, it is still unknown how to construct them in practice. This paper is devoted to the construction of such codes. In particular, two families of linear codes are presented and proved to have good joint spectr...

The problem of finding good linear codes for joint source-channel coding (JSCC) is investigated in this paper. By the code-spectrum approach, it has been proved in the authors' previous paper that a good linear code for the authors' JSCC scheme is a code with a good joint spectrum, so the main task in this paper is to construct linear codes with go...

Cyclic Redundancy Check (CRC) is an established coding method to ensure a low probability of undetected errors in data transmission. CRC is widely used in industrial field bus systems where communication is often executed through different layers. Some layers have their own CRC and add their own specific data to the net data that is meant to be sen...

It is known that a linear two-weight code C over a finite field Fq corresponds both to a multiset in a projective space over Fq that meets every hyperplane in either a or b points for some integers a < b, and to a strongly regular graph whose vertices may be identified with the codewords of C. Here we extend this classical result to the case of a r...

Cyclic Redundancy Check (CRC) is an established coding method to ensure a low probability of undetected errors in data transmission. In CRC, a checksum (Frame Check Sequence, FCS) is attached to the data. The FCS is a result of a polynomial division by a so called generator polynomial. CRC is widely used in industrial communication where the data a...

In this paper, we prove that maximal (k,2)-arcs in projective Hjelmslev planes over chain rings R of nilpotency index 2 exist if and only if .

A (k,n)-arc in the projective Hjelmslev plane PHG(RR3) is defined as a set of k points in the plane such that some n but no n+1 of them are collinear. In this paper, we consider the problem of finding the largest possible size of a (k,n)-arc in PHG(RR3). We present general upper bounds on the size of arcs in the projective Hjelmslev planes over cha...

It is shown that a finite ring R is a Frobenius ring if and only if R(R/RadR) @ Soc(RR)_R(R/\hbox {Rad}\, R)\cong \hbox {Soc}\, (_RR). Other combinatorial characterizations of finite Frobenius rings are presented which have applications in the theory of linear codes over finite rings.

The purpose of this paper is a unified treatment of MacWilliams type identities for several kinds of weight distributions for linear codes over a finite FROBENIUS ring R. The concept of a W-admissible pair of partitions of the ambient space R n is introduced, and MacWilliams type identities are proved for the corresponding weight spectra of linear...

The (k,n)-arcs in projective Hjelmslev plane PHG(RR³) over a finite chain ring R are considered. We prove general upper bounds on the cardinality of such arcs and establish the maximum possible size of the projective (k,n)-arcs with n ∈ {q²,...,q²+q-1}. Constructions of projective arcs in the Hjelmslev planes over the chain rings with 4 and 9 eleme...

The aim of this paper is to develop a theory of linear codes over nite chain rings from a geometric viewpoint. Generalizing a well-known result for lin- ear codes over elds, we prove that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in pro- jective Hjelmslev geometries, in the...

. The results of Constantinescu and Heise [6] concerning homogeneous weight functions on Zm are generalized to nite modules over arbitrary nite rings. Those nite modules M which admit a homogeneous near-weight w : M ! R in the sense of [6] are characterized in terms of the composition factors of their socle. As an application, isometric representat...

In this paper, we consider linear codes over finite chain rings. We present a general mapping which produces codes over smaller alphabets. Under special conditions, these codes are linear over a finite field. We introduce the notion of a linearly representable code and prove that certain MacDonald codes are linearly representable. Finally, we give...

The statement given in the title is proved. Linear codes over chain rings (commutative and noncommutative) are a natural generalization of linear codes over finite fields and of linear codes over integer residue class rings of prime power order. In matters of linear representability there is no obvious reason why we should prefer one chain ring to...

A block code C in F^n is called metrically rigid, if every isometry from C into F^n with respect to theHamming metric is extendable to an isometry of the whole space F^n. The metrical rigidity of some classes of codes is discussed.

As shown by MacWilliams the Hamming weight enumerator of a linear codeC over a finite field can be expressed by the weight enumerator of its dual codeC. A short inductive proof of this formula is given which uses only elementary linear algebra.

Let m and n be positive integers. A map γ:ℤ m →ℝ is called a homogeneous weight, if there is a constant Γ≠0 such that for every nonzero ideal of ℤ m the average weight of its elements is equal to Γ. Generalizing a well-known theorem on the extendability of isometries with respect to the Hamming metric between linear codes over finite fields the aut...