Possible Number Of Codewords Research Articles

Block Codes for Energy-Harvesting Sliding-Window Constrained Channels

Energy-harvesting sliding-window constrained block codes guarantee that within any prescribed window of ℓ consecutive bits the constrained sequence has at least t, t ≥ 1, 1's. Prior art code design methods build upon the finite-state machine description of the (ℓ, t) constraint, but as the number of states equals I choose t, a code design becomes prohibitively complex for mounting I and t. We present a new block code construction that circumvents the enumeration of codewords using a finite-state description of the (ℓ, t)-constraint. The codewords of the block code are encoded and decoded using a single look-up table. For (ℓ = 4, t = 2), the new block codes are maximal, that is, they have the largest possible number of codewords for its parameters.

Two-weight codes: Upper bounds and new optimal constructions

In this paper, we consider q-ary codes of length n and minimum Hamming distance d, which have two weights w1 and w2. These codes are denoted by (n,d,{w1,w2})q codes. Let Aq(n,d,{w1,w2}) denote the largest possible number of codewords in an (n,d,{w1,w2})q code and we simply write A(n,d,{w1,w2}) for A2(n,d,{w1,w2}). Some upper bounds on Aq(n,d,{w1,w2}) are given. The equivalences between binarytwo-weight codes and special combinatorial configurations with certain properties are established and then new upper bounds on A(n,d,{w1,w2}) are derived. For w1,w2∈{2,3,4}, optimal constructions of (n,d,{w1,w2})2 codes are presented. The exact value of A(n,d,{2,3}) is completely determined for all n and d. We determine the exact value of A(n,d,{2,4}) for any positive integer n≡2,4(mod6) and d∈{3,4}. The exact value of A(n,d,{3,4}) is determined for any integer n and d∈{6,7}, or d=5 and n≡0,2,3,11(mod12).

Constructions for optimal cyclic ternary constant-weight codes of weight four and distance six

A cyclic (n,d,w)q code is a q-ary cyclic code of length n, minimum Hamming distance d and weight w. In this paper, we investigate cyclic (n,6,4)3 codes. A new upper bound on CA3(n,6,4), the largest possible number of codewords in a cyclic (n,6,4)3 code, is given. Two new constructions for optimal cyclic (n,6,4)3 codes based on cyclic (n,4,1) difference packings are presented. As a consequence, the exact value of CA3(n,6,4) is determined for any positive integer n≡0,6,18(mod24). We also obtain some other infinite classes of optimal cyclic (n,6,4)3 codes.

Constructions of cyclic quaternary constant-weight codes of weight three and distance four

A cyclic $$(n,d,w)_q$$ code is a cyclic q-ary code of length n, constant-weight w and minimum distance d. A cyclic $$(n,d,w)_q$$ code with the largest possible number of codewords is said to be optimal. Optimal nonbinary cyclic $$(n,d,w)_q$$ codes were first studied in our recent paper (Lan et al. in IEEE Trans Inf Theory 62(11):6328–6341, 2016). In this paper, we continue to discuss the constructions of optimal cyclic $$(n,4,3)_q$$ codes. We establish the connection between cyclic $$(n,4,3)_{q}$$ codes and $$q-1$$ mutually orbit-disjoint cyclic (n, 3, 1) difference packings (briefly (n, 3, 1)-CDPs). For the case of $$q=4$$ , we construct three mutually orbit-disjoint (n, 3, 1)-CDPs by constructing a pair of strongly orbit-disjoint (n, 3, 1)-CDPs, which are obtained from Skolem-type sequences. As a consequence, we completely determine the number of codewords of an optimal cyclic $$(n,4,3)_{4}$$ code.

$(m\kern -1pt,n\kern -1pt,3\kern -1pt,1)$ Optical Orthogonal Signature Pattern Codes With Maximum Possible Size

Kitayama proposed a novel code-division multiple-access (CDMA) network for image transmission called spatial CDMA. Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial CDMA. An (m,n,k,λ)-OOSPC is a set of m × n (0,1)-matrices with Hamming weight k and maximum correlation value λ. Let Θ (m,n,k,λ) be the largest possible number of codewords among all (m,n,k,λ) -OOSPCs. In this paper, we concentrate on the calculation of the exact value of Θ (m,n,3,1) and the construction of an (m,n,3,1)-OOSPC with Θ (m,n,3,1) codewords. As a consequence, we show that Θ (m,n,3,1)=[ mn-1/6]-1 when mn≡ 14, 20(mod 24), or mn≡ 8, 16(mod 24) and gcd (m,n,4)=2 , or mn≡ 2(mod 6) and gcd (m,n,4)=4 , and Θ (m,n,3,1)= [mn-1/6] otherwise.

Further results on optimal (<italic>m</italic>, <italic>n</italic>, 4, 1) optical orthogonal signature pattern codes

To support high-speed multiple access network applications, especially image applications, Kitayama rst presented the concept of optical orthogonal signature pattern code (OOSPC). An (<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ)-OOSPC is a family of (0, 1)-matrices of Hamming weight <italic>k</italic> satisfying two correlation properties. Let Θ(<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ) denote the largest possible number of codewords among all (<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ)-OOSPCs. An (<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ)-OOSPC with Θ(<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ) codewords is said to be optimal. In this paper, we confirm Θ(<italic>m</italic>, <italic>n</italic>, 4, 1) = 「<italic>mn</italic>-1/12」, in other words, we construct an (<italic>m</italic>, <italic>n</italic>, 4, 1)-OOSPC with 「<italic>mn</italic>-1/12」 codewords for any positive integers m and n satisfying one of the following: (1) <italic>mn</italic>≡ 8; 16 (mod 24), each of whose odd prime factor is congruent to 1 modulo 6, with the exception of gcd(<italic>m</italic>, <italic>n</italic>, 2) = 1, or <italic>mn</italic>≡ 16 (mod 32) and gcd(<italic>m</italic>, <italic>n</italic>, 4) = 2; (2) <italic>mn</italic>≡ 0 (mod 24) and gcd(<italic>m</italic>, <italic>n</italic>, 6) = 2.

Improved Semidefinite Programming Bound on Sizes of Codes

Let A(n,d) (respectively A(n,d,w)) be the maximum possible number of codewords in a binary code (respectively, binary constant-weight w code) of length n and minimum Hamming distance at least d. By adding new linear constraints to Schrijver's semidefinite programming bound, which is obtained from block-diagonalizing the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on A(n,d), namely A(18,8) ≤ 71 and A(19,8) ≤ 131. Twenty three new upper bounds on A(n,d,w) for n ≤ 28 are also obtained by a similar way.

Combinatorial constructions for maximum optical orthogonal signature pattern codes

An (m,n,k,λa,λc) optical orthogonal signature pattern code (OOSPC) is a family C of m×n (0,1)-matrices of Hamming weight k satisfying two correlation properties. OOSPCs find application in transmitting two-dimensional image through multicore fiber in CDMA networks. Let Θ(m,n,k,λa,λc) denote the largest possible number of codewords among all (m,n,k,λa,λc)-OOSPCs. An (m,n,k,λa,λc)-OOSPC with Θ(m,n,k,λa,λc) codewords is said to be maximum. For the case λa=λc=λ, the notations (m,n,k,λa,λc)-OOSPC and Θ(m,n,k,λa,λc) can be briefly written as (m,n,k,λ)-OOSPC and Θ(m,n,k,λ). In this paper, some direct constructions for (3,n,4,1)-OOSPCs, which are based on skew starters and an application of the Theorem of Weil on multiplicative character sums, are given for some positive integer n. Several recursive constructions for (m,n,k,1)-OOSPCs are presented by means of incomplete different matrices and group divisible designs. By utilizing those constructions, the number of the codewords of a maximum (m,n,4,1)-OOSPC is determined for any positive integers m,n such that gcd(m,18)=3 and n≡0(mod12). It is established that Θ(m,n,4,1)=(mn−12)/12 for any positive integers m,n such that gcd(m,18)=3 and n≡0(mod12).

Maximum two-dimensional (u × v, 4, 1, 3)-OOCs

Let Φ(u × v, k, λa, λc) be the largest possible number of codewords among all two-dimensional (u × v, k, λa, λc) optical orthogonal codes. A 2-D (u × v, k, λa, λc)-OOC with Φ(u × v, k, λa, λc) codewords is said to be maximum. In this paper, the number of codewords of a maximum 2-D (u × v, 4, 1, 3)-OOC has been determined.

A New Class of Binary Constant Weight Codes Derived by Groups of Linear Fractional Mappings

Let A(n, d, w) denote the maximum possible number of code words in binary (n,d,w) constant weight codes. For smaller instances of (n, d, w)s, many improvements have occurred over the decades. However, unknown instances still remain for larger (n, d, w)s (for example, those of n > 30 and d > 10). In this paper, we propose a new class of binary constant weight codes that fill in the remaining blank instances of (n, d, w)s. Specifically, we establish several new non-trivial lower bounds such as 336 for A(64, 12, 8), etc. (listed in Table 2). To obtain these results, we have developed a new systematic technique for construction by means of groups acting on some sets. The new technique is performed by considering a triad (G, f) := (Group G, Set Ω, Action f on Ω) simultaneously. Our results described in Sect. 3 are obtained by using permutations of the elements of a set that include ∞ homogeneously like the other elements, which play a role to improve their randomness. Specifically, in our examples, we adopt the following model such as (PGL2( q), 1( q), linear fractional action of subgroups of PGL2( q) on 1( q)) as a typical construction model. Moreover, as an application, the essential examples in [7] constructed by using an alternating group are again reconstructed with our new technique of a triad model, after which they are all systematically understood in the context of finite subgroups that act fractionally on a projective space over a finite field.

Improved upper bounds on sizes of codes

Let A(n,d) denote the maximum possible number of codewords in a binary code of length n and minimum Hamming distance d. For large values of n, the best known upper bound, for fixed d, is the Johnson bound. We give a new upper bound which is at least as good as the Johnson bound for all values of n and d, and for each d there are infinitely many values of n for which the new bound is better than the Johnson bound. For small values of n and d, the best known method to obtain upper bounds on A(n,d) is linear programming. We give new inequalities for the linear programming and show that with these new inequalities some of the known bounds on A(n,d) for n/spl les/28 are improved.

A table of upper bounds for binary codes

Let A(n, d) denote the maximum possible number of codewords in an (n, d) binary code. We establish four new bounds on A(n, d), namely, A(21, 4)/spl les/43689, A(22, 4)/spl les/87378, A(22, 6)/spl les/6941, and A(23, 4)/spl les/173491. Furthermore, using previous upper bounds on the size of constant-weight binary codes, we reapply known methods to generate a table of bounds on A(n, d) for all n/spl les/28. This table extends the range of parameters compared with previously known tables.

Upper bounds for constant-weight codes

Let A(n,d,w) denote the maximum possible number of codewords in an (n,d,w) constant-weight binary code. We improve upon the best known upper bounds on A(n,d,w) in numerous instances for n/spl les/24 and d/spl les/12, which is the parameter range of existing tables. Most improvements occur for d=8, 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n/spl les/28 and d/spl les/14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n,d,w) by means of mapping constant-weight codes into Euclidean space. This approach produces, among other results, a bound on A(n,d,w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doubly-constant-weight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doubly-bounded-weight codes, which may be thought of as a generalization of the doubly-constant-weight codes. Subsequently, a class of Euclidean-space codes, called zonal codes, is introduced, and a bound on the size of such codes is established. This is used to derive bounds for doubly-bounded-weight codes, which are in turn used to derive bounds on A(n,d,w). We also develop a universal method to establish constraints that augment the Delsarte inequalities for constant-weight codes, used in the linear programming bound. In addition, we present a detailed survey of known upper bounds for constant-weight codes, and sharpen these bounds in several cases. All these bounds, along with all known dependencies among them, are then combined in a coherent framework that is amenable to analysis by computer. This improves the bounds on A(n,d,w) even further for a large number of instances of n, d, and w.

Multidimensional stochastic matrices and error-correcting codes

The exact value of τ( n, q, r), the largest possible number of codewords in a code of length n, distance r+1, where the entries of the codewords are members of an alphabet of q elements, is determined in large number of cases. Also, a new upper bound is given for this function τ. These exact values and estimations are used to answer a question of Jurkat and Ryser concerning the existence of higher dimensional stochastic (0, 1) matrices in several cases.

A class of optimum nonlinear double-error-correcting codes

Nonlinear double-error-correcting block codes of length (2 n − 1) ( n even) are presented in this paper. They have the largest possible number of code-words for their length and minimum distance and are formed by adjoining to a certain linear code (referred to as the “kernel”) a specific subset of its cosets. The kernel results from the juxtaposition and superposition of Bose-Chaudhuri-Hocquenghem codes of length (2 n −1 − 1). The presented codes are systematic and are comparable to the corresponding linear codes with regard to the complexity of the encoding and decoding operations.