On transitive partitions of an <Emphasis Type="Italic">n</Emphasis>-cube into codes

We present methods to construct transitive partitions of the set E ⁿ of all binary vectors of length n into codes. In particular, we show that for all n = 2 ^k ? 1, k ≥ 3, there exist transitive partitions of E ⁿ into perfect transitive codes of length n.     

On partitions of an <Emphasis Type="Italic">n</Emphasis>-cube into nonequivalent perfect codes

We prove that for all n = 2^k ? 1, k ≥ 5, there exists a partition of the set of all binary vectors of length n into pairwise nonequivalent perfect binary codes of length n with distance 3.     

Uniqueness of some optimal superimposed codes

Four new results on the uniqueness of optimal superimposed codes are presented, namely, the uniqueness of (w, r) superimposed codes of size N × T with \(N = \left( {\begin{array}{*{20}c} {w + r + 1} \\ w \\ \end{array} } \right)\) and T = w + r + 1 and the uniqueness of (2, 2) superimposed codes of size 18 × 9, (2, 2) superimposed codes of size 14 × 7, and (3, 3) superimposed codes of size 66 × 11.     

Sufficient Conditions for Monotonicity of the Undetected Error Probability for Large Channel Error Probabilities

The performance of a linear error-detecting code in a symmetric memoryless channel is characterized by its probability of undetected error, which is a function of the channel symbol error probability, involving basic parameters of a code and its weight distribution. However, the code weight distribution is known for relatively few codes since its computation is an NP-hard problem. It should therefore be useful to have criteria for properness and goodness in error detection that do not involve the code weight distribution. In this work we give two such criteria. We show that a binary linear code C of length n and its dual code C^⊥ of minimum code distance d^⊥ are proper for error detection whenever d^⊥ ≥ ?n/2? + 1, and that C is proper in the interval [(n + 1 ? 2d^⊥)/(n ? d^⊥); 1/2] whenever ?n/3? + 1 ≤ d^⊥ ≤ ?n/2?. We also provide examples, mostly of Griesmer codes and their duals, that satisfy the above conditions.     

Propelinear codes related to some classes of optimal codes

A code is said to be propelinear if its automorphism group contains a subgroup that acts regularly on codewords. We show propelinearity of complements of cyclic codes C _1,i , (i, 2 ^m ? 1) = 1, of length n = 2 ^m ? 1, including the primitive two-error-correcting BCH code, to the Hamming code; the Preparata code to the Hamming code; the Goethals code to the Preparata code; and the Z₄-linear Preparata code to the Z₄-linear perfect code.     

MDS codes in Doob graphs

The Doob graph D(m, n), where m > 0, is a Cartesian product of m copies of the Shrikhande graph and n copies of the complete graph K ₄ on four vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). We give a characterization of MDS codes in Doob graphs D(m, n) with code distance at least 3. Up to equivalence, there are m ³/36+7m ²/24+11m/12+1?(m mod 2)/8?(m mod 3)/9 MDS codes with code distance 2m + n in D(m, n), two codes with distance 3 in each of D(2, 0) and D(2, 1) and with distance 4 in D(2, 1), and one code with distance 3 in each of D(1, 2) and D(1, 3) and with distance 4 in each of D(1, 3) and D(2, 2).     

On the Construction of Transitive Codes

Application of some known methods of code construction (such as the Vasil'ev, Plotkin, and Mollard methods) to transitive codes satisfying certain auxiliary conditions yields infinite classes of large-length transitive codes, in particular, at least ?k/2?² nonequivalent perfect transitive codes of length n = 2 ^k ? 1, k > 4. A similar result is valid for extended perfect transitive codes.     

On weak isometries of Preparata codes

Two codes C ₁ and C ₂ are said to be weakly isometric if there exists a mapping J: C ₁ → C ₂ such that for all x, y in C ₁ the equality d(x, y) = d holds if and only if d(J(x), J(y)) = d, where d is the code distance of C ₁. We prove that Preparata codes of length n ≥ 2¹² are weakly isometric if and only if the codes are equivalent. A similar result is proved for punctured Preparata codes of length at least 2¹⁰ ? 1.     

Generalized Preparata codes and 2-resolvable Steiner quadruple systems

We consider generalized Preparata codes with a noncommutative group operation. These codes are shown to induce new partitions of Hamming codes into cosets of these Preparata codes. The constructed partitions induce 2-resolvable Steiner quadruple systems S(n, 4, 3) (i.e., systems S(n, 4, 3) that can be partitioned into disjoint Steiner systems S(n, 4, 2)). The obtained partitions of systems S(n, 4, 3) into systems S(n, 4, 2) are not equivalent to such partitions previously known.     

A Note on the Uniqueness of (<Emphasis Type="Italic">w,r</Emphasis>) Cover-Free Codes

A binary code is called a (w, r) cover-free code if it is the incidence matrix of a family of sets where the intersection of any w sets is not covered by the union of any other r sets. For certain (w, r) cover-free codes with a simple structure, we obtain a new condition of optimality and uniqueness up to row and/or column permutations.     

On the symmetry group of the Mollard code

We study the symmetry group of a binary perfect Mollard code M(C,D) of length tm + t + m containing as its subcodes the codes C ¹ and D ² formed from perfect codes C and D of lengths t and m, respectively, by adding an appropriate number of zeros. For the Mollard codes, we generalize the result obtained in [1] for the symmetry group of Vasil’ev codes; namely, we describe the stabilizer

$$Sta{b_{{D^2}}}$$

Sym(M(C,D)) of the subcode D ² in the symmetry group of the code M(C,D) (with the trivial function). Thus we obtain a new lower bound on the order of the symmetry group of the Mollard code. A similar result is established for the automorphism group of Steiner triple systems obtained by the Mollard construction but not necessarily associated with perfect codes. To obtain this result, we essentially use the notions of “linearity” of coordinate positions (points) of a nonlinear perfect code and a nonprojective Steiner triple system.

     

Polar Codes with Higher-Order Memory

We introduce a construction of a set of code sequences {C_n^(m) : n ≥ 1, m ≥ 1} with memory order m and code length N(n). {C_n^(m)} is a generalization of polar codes presented by Ar?kan in [1], where the encoder mapping with length N(n) is obtained recursively from the encoder mappings with lengths N(n ? 1) and N(n ? m), and {C_n^(m)} coincides with the original polar codes when m = 1. We show that {C_n^(m)} achieves the symmetric capacity I(W) of an arbitrary binary-input, discrete-output memoryless channel W for any fixed m. We also obtain an upper bound on the probability of block-decoding error P_e of {C_n^(m)} and show that \({P_e} = O({2^{ - {N^\beta }}})\) is achievable for β < 1/[1+m(? ? 1)], where ? ∈ (1, 2] is the largest real root of the polynomial F(m, ρ) = ρ^m ? ρ^{m ? 1} ? 1. The encoding and decoding complexities of {C_n^(m)} decrease with increasing m, which proves the existence of new polar coding schemes that have lower complexity than Ar?kan’s construction.     

New quantum codes from dual-containing cyclic codes over finite rings

Let \(R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}\), where \(\mathbb {F}_{2^{m}}\) is the finite field with \(2^{m}\) elements, m is a positive integer, and u is an indeterminate with \(u^{k+1}=0.\) In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of \(2^{m}\)-ary quantum codes is obtained via the Gray map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R.     

On Cosets of Weight 4 of Binary BCH Codes with Minimum Distance 8 and Exponential Sums

We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length n = 2 ^m (m odd) with minimum distance 8. In the previous paper [1], we described coset weight distributions of such codes for cosets of weight j = 1, 2, 3, 5, 6. Here we obtain exact expressions for the number of codewords of weight 4 in terms of exponential sums of three types, in particular, cubic sums and Kloosterman sums. This allows us to find the coset distribution of binary primitive (narrow-sense) BCH codes with minimum distance 8 and also to obtain some new results on Kloosterman sums over finite fields of characteristic 2.     

The weight-constrained maximum-density subtree problem and related problems in trees

Given a tree T=(V,E) of n nodes such that each node v is associated with a value-weight pair (val _v ,w _v ), where value val _v is a real number and weight w _v is a non-negative integer, the density of T is defined as \(\frac{\sum_{v\in V}{\mathit{val}}_{v}}{\sum_{v\in V}w_{v}}\). A subtree of T is a connected subgraph (V′,E′) of T, where V′?V and E′?E. Given two integers w _min? and w _max?, the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T′=(V′,E′) satisfying w _min?≤∑_v∈V′ w _v ≤w _max?. In this paper, we first present an O(w _max? n)-time algorithm to find a weight-constrained maximum-density path in a tree T, and then present an O(w _max? ² n)-time algorithm to find a weight-constrained maximum-density subtree in T. Finally, given a node subset S?V, we also present an O(w _max? ² n)-time algorithm to find a weight-constrained maximum-density subtree in T which covers all the nodes in S.     

List decoding of the first-order binary Reed-Muller codes

A list decoding algorithm is designed for the first-order binary Reed-Muller codes of length n that reconstructs all codewords located within the ball of radius n/2(1 ? ?) about the received vector and has the complexity of O(n ln²(min{? ^?2, n})) binary operations.     

Vectorizing computations at decoding of nonbinary codes with small density of checks

A modification of the decoding q-ary Sum Product Algorithm (q-SPA) was proposed for the nonbinary codes with small check density based on the permutation matrices. The algorithm described has a vector realization and operates over the vectors defined on the field GF(q), rather than over individual symbols. Under certain code parameters, this approach enables significant speedup of modeling.     

On new completely regular <Emphasis Type="Italic">q</Emphasis>-ary codes

In this paper, new completely regular q-ary codes are constructed from q-ary perfect codes. In particular, several new ternary completely regular codes are obtained from the ternary [11, 6, 5] Golay code. One of these codes with parameters [11, 5, 6] has covering radius ρ = 5 and intersection array (22, 20, 18, 2, 1; 1, 2, 9, 20, 22). This code is dual to the ternary perfect [11, 6, 5] Golay code. Another [10, 5, 5] code has covering radius ρ = 4 and intersection array (20, 18, 4, 1; 1, 2, 18, 20). This code is obtained by deleting one position of the former code. All together, the ternary Golay code results in eight completely regular codes, only four of which were previously known. Also, new infinite families of completely regular codes are constructed from q-ary Hamming codes.     

On ℤ<Subscript>4</Subscript>-linear codes with the parameters of Reed-Muller codes

For any pair of integers r and m, 0 ≤ r ≤ m, we construct a class of quaternary linear codes whose binary images under the Gray map are codes with the parameters of the classical rth-order Reed-Muller code RM(r, m).     

Quantum codes from cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$

In this paper, we study quantum codes over \(F_q\) from cyclic codes over \(F_q+uF_q+vF_q+uvF_q,\) where \(u^2=u,~v^2=v,~uv=vu,~q=p^m\), and p is an odd prime. We give the structure of cyclic codes over \(F_q+uF_q+vF_q+uvF_q\) and obtain self-orthogonal codes over \(F_q\) as Gray images of linear and cyclic codes over \(F_q+uF_q+vF_q+uvF_q\). In particular, we decompose a cyclic code over \(F_q+uF_q+vF_q+uvF_q\) into four cyclic codes over \(F_q\) to determine the parameters of the corresponding quantum code.