Perfect Codes Research Articles

The influence of non-perfect code subgroups on the structure of groups

Let [Formula: see text] be a graph. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when [Formula: see text] is an independent set and every vertex of [Formula: see text] is adjacent to exactly one vertex in [Formula: see text]. Let [Formula: see text] = Cay(G, S) be a Cayley graph of a finite group [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when there exists a Cayley graph [Formula: see text] of [Formula: see text] such that [Formula: see text] is a perfect code of [Formula: see text]. Recently, groups whose set of all subgroup perfect codes forms a chain are classified. Also, groups with no proper nontrivial subgroup perfect code are characterized. In this paper, we generalize it and classify groups whose set of all non-perfect code subgroups forms a chain.

Perfect codes in 2-valent Cayley digraphs on abelian groups

For a digraph Γ, a subset C of V(Γ) is a perfect code if C is a dominating set such that every vertex of Γ is dominated by exactly one vertex in C. In this paper, we classify strongly connected 2-valent Cayley digraphs on abelian groups admitting a perfect code, and determine completely all perfect codes of such digraphs.

Almost Perfect Linear Lee Codes of Packing Radius 2 Only Exist for Small Dimensions

Almost Perfect Linear Lee Codes of Packing Radius 2 Only Exist for Small Dimensions

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e≥3, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q>3. In the 1970s, this was proved for q a prime power, for q=2r3s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q.

Subgroup total perfect codes in Cayley sum graphs

Subgroup total perfect codes in Cayley sum graphs

Perfect directed codes in Cayley digraphs

<p>A perfect directed code (or an efficient twin domination) of a digraph is a vertex subset where every other vertex in the digraph has a unique in- and a unique out-neighbor in the subset. In this paper, we show that a digraph covers a complete digraph if and only if the vertex set of this digraph can be partitioned into perfect directed codes. Equivalent conditions for subsets in Cayley digraphs to be perfect directed codes are given. Especially, equivalent conditions for normal subsets, normal subgroups, and subgroups to be perfect directed codes in Cayley digraphs are given. Moreover, we show that every subgroup of a finite group is a perfect directed code for a transversal Cayley digraph.</p>

On connected components and perfect codes of proper order graphs of finite groups

On connected components and perfect codes of proper order graphs of finite groups

The Perfect Codes of Non-coprime and Coprime Graphs

In this paper, we focus on the perfect and total perfect codes of the non-coprime and coprime graphs associated to the dihedral groups and finite Abelian groups. We used the advantage of independent sets and tried to present the independent polynomial for them.

Constructions and invariants of optimal codes in the Lee metric

We propose concatenated and switching methods for the construction of single-error-correcting perfect and diameter codes in the Lee metric. We analyze ranks and kernels of diameter perfect codes obtained by the switching construction.

Perfect codes in two-dimensional algebraic lattices

In the present paper, we investigate the existence of lattice perfect codes in two families of two-dimensional lattices built from the Minkowski embedding of the ring of integers of a quadratic field K=Q[l], where l is a square-free integer. We analyzed the forms and quantities of perfect codes for two of these families lattices, one Orthogonal and the other Non-Orthogonal and we verified that we always managed to find a perfect code in these two families of lattices. The greater the value of l the more perfect codes we get.

On Almost Perfect Linear Lee Codes of Packing Radius 2

More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee codes <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> of packing radius <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> in Zn for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> ≥ 2 and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≥ 3. Recently, Leung and the second author proved that if <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> is linear, then the Golomb-Welch conjecture is valid for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> = 2 and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≥ 3. In this paper, we consider the classification of linear Lee codes with the second-best possibility, that is the density of the lattice packing of Zn by Lee spheres <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, r</i> ) equals | <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,r</i> )|/| <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,r</i> )|+1 . We show that, for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> = 2 and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≡ 0, 3, 4 (mod 6), this packing density can never be achieved.

On linear diameter perfect Lee codes with distance 6

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius r≥2 and dimension n≥3. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (2011) [5] proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with distance greater than four besides the DPL(3,6) code? Later, Horak and AlBdaiwi (2012) [12] conjectured that there are no DPL(n,d) codes for dimension n≥3 and distance d>4 except for (n,d)=(3,6). In this paper, we give a counterexample to this conjecture. Moreover, we prove that for n≥3, there is a linear DPL(n,6) code if and only if n=3,11.

Perfect codes in proper intersection power graphs of finite groups

Perfect codes in proper intersection power graphs of finite groups

Quasi-Cyclic Perfect Codes in Doob Graphs and Special Partitions of Galois Rings

The Galois ring GR$(4^\Delta)$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $\Delta$ over $Z_4$. For any odd $\Delta$ larger than $1$, we construct a partition of GR$(4^\Delta) \backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^\Delta)$ and, if $\Delta$ is not a multiple of $3$, under the action of the automorphism group of GR$(4^\Delta)$. As a corollary, this implies the existence of quasi-cyclic additive $1$-perfect codes of index $(2^\Delta-1)$ in $D((2^\Delta-1)(2^\Delta-2)/{6}, 2^\Delta-1 )$ where $D(m,n)$ is the Doob metric scheme on $Z^{2m+n}$.

A family of diameter perfect constant-weight codes from Steiner systems

If S is a transitive metric space, then |C|⋅|A|≤|S| for any distance-d code C and a set A, “anticode”, of diameter less than d. For every Steiner S(t,k,n) system S, we show the existence of a q-ary constant-weight code C of length n, weight k (or n−k), and distance d=2k−t+1 (respectively, d=n−t+1) and an anticode A of diameter d−1 such that the pair (C,A) attains the code–anticode bound and the supports of the codewords of C are the blocks of S (respectively, the complements of the blocks of S). We study the problem of estimating the minimum value of q for which such a code exists, and find that minimum for small values of t.

Block codes on pomset metric

Given a regular multiset [Formula: see text] on [Formula: see text], a partial order [Formula: see text] on [Formula: see text], and a label map [Formula: see text] defined by [Formula: see text] with [Formula: see text], we define a pomset block metric [Formula: see text] on the direct sum [Formula: see text] of [Formula: see text] based on the pomset [Formula: see text]. The pomset block metric extends the classical pomset metric introduced by Sudha and Selvaraj and generalizes the poset block metric introduced by Alves et al. over [Formula: see text]. The space [Formula: see text] is called the pomset block space and we determine the complete weight distribution of it. Further, [Formula: see text]-perfect pomset block codes for ideals with partial and full counts are described. Then, for block codes with chain pomset, the packing radius and Singleton bound are established. The relation between maximum distance separable (MDS) codes and [Formula: see text]-perfect codes for any ideal [Formula: see text] is investigated. Moreover, the duality theorem for an MDS pomset block code is established when all the blocks have the same size.

Projective tilings and full-rank perfect codes

A tiling of a vector space S is the pair (U, V) of its subsets such that every vector in S is uniquely represented as the sum of a vector from U and a vector from V. A tiling is connected to a perfect codes if one of the sets, say U, is projective, i.e., the union of one-dimensional subspaces of S. A tiling (U, V) is full-rank if the affine span of each of U, V is S. For finite non-binary vector spaces of dimension at least 6 (at least 10), we construct full-rank tilings (U, V) with projective U (both U and V, respectively). In particular, that construction gives a full-rank ternary 1-perfect code of length 13, solving a known problem. We also discuss the treatment of tilings with projective components as factorizations of projective spaces.

Characterizing subgroup perfect codes by 2-subgroups

A perfect code in a graph $$\Gamma $$ is a subset C of $$V(\Gamma )$$ such that no two vertices in C are adjacent and every vertex in $$V(\Gamma ){\setminus } C$$ is adjacent to exactly one vertex in C. Let G be a finite group and C a subset of G. Then C is said to be a perfect code of G if there exists a Cayley graph of G admiting C as a perfect code. It is proved that a subgroup H of G is a perfect code of G if and only if a Sylow 2-subgroup of H is a perfect code of G. This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of 2-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups $$\textrm{PSL}(2,q)$$ is given.

An enumeration of 1-perfect ternary codes

We study codes with parameters of the ternary Hamming (n=(3m−1)/2,3n−m,3) code, i.e., ternary 1-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary 1-perfect codes of rank n−m+1, count their number, and prove that all such codes can be obtained from each other by a sequence of two-coordinate switchings. We enumerate ternary 1-perfect codes of length 13 obtained by concatenation from codes of lengths 9 and 4; we find that there are 93241327 equivalence classes of such codes.

Beating the Probabilistic Lower Bound on q-Perfect Hashing

In this paper, we investigate the achievable rate of perfect hash code. For an integer $$q\ge 2$$ , a perfect q-hash code C is a block code over $$[q]:=\{1,\ldots ,q\}$$ of length n in which every subset $$\{{\textbf{c}}_1,{\textbf{c}}_2,\dots ,{\textbf{c}}_q\}$$ of q elements is separated, i.e., there exists $$i\in [n]$$ such that $$\{\textrm{proj}_i({\textbf{c}}_1),\dots ,\textrm{proj}_i({\textbf{c}}_q)\}=[q]$$ , where $$\textrm{proj}_i({\textbf{c}}_j)$$ denotes the ith position of $${\textbf{c}}_j$$ . Finding the maximum size M(n, q) of perfect q-hash codes of length n, for given q and n, is a fundamental problem in combinatorics, information theory, and computer science. In this paper, we are interested in asymptotic behavior of this problem. Precisely speaking, we will focus on the quantity $$R_q:=\limsup _{n\rightarrow \infty }\frac{\log _2 M(n,q)}{n}$$ . A well-known probabilistic argument shows an existence lower bound on $$R_q$$ , namely $$R_q\ge \frac{1}{q-1}\log _2\left( \frac{1}{1-q!/q^q}\right) $$ (Fredman and Komlós, SIAM J Algebr Discrete Methods 5(1):61–68, 1984; Körner, SIAM J Algebr Discrete Methods 7(4):560–570, 1986). This is still the best-known lower bound till now except for the case $$q=3$$ (Körner and Marton, Eur J Comb 9(6):523–530, 1988). The improved lower bound of $$R_3$$ was discovered in 1988 and there has been no progress on the lower bound of $$R_q$$ for more than 30 years. In this paper we show that this probabilistic lower bound can be improved for q from 4 to 15 and all odd integers between 17 and 25, and all sufficiently large q.