Optimal binary codes from one-lee weight codes and two-lee weight projective codes over ℤ4

<p style='text-indent:20px;'>Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.</p>

A coordinate of a binary code of size M is said to be balanced if the number of zero and ones in the coordinate is either \(\lfloor M/2\rfloor \) or \(\lceil M/2 \rceil \) (that is, exactly \(M/2\) for even M). Since good codes (of various types) tend to be balanced in all coordinates, various conjectures have been made regarding the existence of such codes. It is here shown that there are parameters for which there are no optimal binary error-correcting codes with a balanced coordinate. This is proved by the code attaining \(A(17,8) = 36\), which is shown to be unique here; \(A(n,d)\) denotes the maximum size of a binary code of length n and minimum distance d. It is further shown that \(A(18,8) \le 68\).

A strategy is described which allowed new optimal binary linear codes to be constructed. Using the method, among other codes a new [48, 16, 16] optimal linear binary block code was constructed. The construction of this new code settles a conjecture mentioned in the paper by Janosov and colleagues (see appropriate Reference in Letter) where the authors found a new optimal [47, 15, 16]‐code, whose parameters are also relevant to the present author's new method.

Optimal binary linear codes of length ⩽30

We introduce linear cyclic codes over the ring F/sub 2/+uF/sub 2/={0,1,u,u~=u+1}, where u/sup 2/=0 and study them by analogy with the Z/sub 4/ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z/sub 4/-codes. Unlike the Z/sub 4/ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map.

Optimal binary linear codes of dimension at most seven

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ were reported in the literature, while vast knowledge exists on the ones with $$w_{\min }/w_{\max }> 1/2$$ . Herein, $$w_{\min }$$ and $$w_{\max }$$ respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established.

Some new restrictions on the last element of a distance spectrum for optimal binary block codes are presented.

A new family of nonlinear codes, called weak flip codes, is presented and is shown to contain many beautiful properties. In particular, the subfamily of fair weak flip codes can be seen as a generalization of linear codes. Different from linear codes that only exist for a number of codewords M being an integer-power of 2, the fair weak flip code can be defined for an arbitrary M. It is then noted that the fair weak flip codes are related to binary nonlinear Hadamard codes: both code families maximize the minimum Hamming distance and meet the Plotkin bound. However, while the binary nonlinear Hadamard codes have only been shown to possess good Hamming-distance properties, the fair weak flip codes are proven to be globally optimal (in the sense of minimizing the error probability) among all linear or nonlinear codes for the binary erasure channel (BEC) for many values of the blocklength n and for M≤6. For M>6, similar optimality results are conjectured. The results in this paper are founded on a new powerful tool for the analysis and generation of block codes: the column-wise approach to the codebook matrix.

New metrics and distances for linear codes over the ring Fq[u]/(u) are defined, which generalize the Gray map, Lee weight, and Bachoc weight; and new bounds on distances are given. Two characterizations of self-dual codes over Fq[u]/(u) are determined in terms of linear codes over Fq. An algorithm to produce such self-dual codes is also established. 1.Introduction. Many optimal codes have been obtained by studying codes over general rings rather than fields. Lately, codes over finite chain rings (of which Fq[u]/(u) is an example) have been a source of many interesting properties [6, 8, 3]. Gulliver and Harada [4] found good examples of ternary codes over F3 using a particular type of Gray map. Siap and Ray-Chaudhuri in [9] established a relation between codes over Fq[u]/(u − a) and codes over Fq, which was used to obtained new codes over F3 and F5. In this paper we present a certain generalization of the method used in [4] and [9], defining a family of metrics for linear codes over Fq[u]/(u) and obtaining as particular examples the Gray map, the Gray weight, the Lee weight and the Bachoc weight. For the latter, we give a new bound on the distance of those codes. It also shows that the Gray images of codes over F2 + uF2 are more powerful than codes obtained by the so-called u—u+v condition. With these tools in hand, we study conditions for self-duality of codes over Fq[u]/(u). In [7] the authors study the case of self-dual cyclic codes in terms of the generator polynomials. In this paper we study self-dual codes in terms of linear codes over Fq that are obtained as images under the maps defined on the first part of the paper. We provide a way to construct many self-dual codes over Fq starting from a self-dual code over Fq[u]/(u). We also study self-dual codes in terms of the torsion codes, and provide a way to construct many self-dual codes over Fq[u]/(u) starting from a self-orthogonal code over Fq. Our results contain many of the properties studied by Bachoc for self-dual codes over F3 + uF3 in [1]. 2. Metric for Codes over Fq[u]/(u). We will use R(q, t) to denote the commutative ring Fq[u]/(u). The q elements of this ring can be represented in two different forms, and we will use the most appropriate in each case. First, we can use the ∗The project was partially supported by Office of Research of the University of Michigan-Flint. †2000 Mathematics Subject Classification: Primary: 94B05, 94B60, Secondary: 11T71. ‡

In this paper, we consider the optimal design of linear space codes for an indoor multiple input multiple output (MIMO) visible light communication (VLC) system with maximum-likelihood (ML) detection, where we assume that the full knowledge of channel state information (CSI) is available at both the transmitter and the receiver. For such a system with on-off-keying (OOK) modulation, an optimal closed-form linear space code that maximizes the minimum Euclidean distance for ML detection is attained for the scenario with two transmitters and multiple receivers. Our results reveal the fact that when the channel is good, the optimal code has either a diagonal or an anti-diagonal structure, and when the channel is bad, the optimal code is to diagonalize the channel matrix first, then, optimally distribute the total transmission power into each eigen-subchannel, and then, follow a repetition coding (RC) scheme. Computer simulations verify both the accuracy and the validity of our theoretical analysis, and show that our newly proposed linear space code provides significant error performance gains over currently available space modlution (SM) and RC in literature without CSI at the transmitter.

Cyclic codes are important in theory, as they are closely related to a number of areas of mathematics. Cyclic codes are also important in practice, as they have efficient encoding and decoding algorithms. An infinite family of cyclic codes over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) is said to have linearly-best-known parameters if for any [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ] code <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> in this family, there is no known [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′] linear code over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) such that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′ > <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> . An infinite family of cyclic codes over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) is said to have cyclicly-best-known parameters if for any [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ] code <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> in this family, there is no known [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′] cyclic code over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) such that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′ > <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> . It is very rare to see an infinite family of binary cyclic codes with cyclicly-best-known parameters whose duals codes have also cyclicly-best-known parameters. The objective of this paper is to study such family of binary cyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>m</i></sup> – 1 and dimension 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>m</i></sup> – 1 – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> – 1)/2, denoted by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> , and their dual codes <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> . The weight distribution of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> is settled and the parameters of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> are investigated in this paper. A larger family of binary cyclic codes <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,<i>r</i>)</sub> and their duals are also constructed and studied in this paper, where 0 ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> – 1.

The design of space-time codes for frequency flat, spatially correlated MIMO fading channels is considered. The focus of the paper is on the class of space-time block codes known as linear dispersion (LD) codes, introduced by Hassibi and Hochwald. The LD codes are optimized with respect to the mutual information between the inputs to the space-time encoder and the output of the channel. The use of the mutual information as both a design criterion and a performance measure is justified by allowing soft decisions at the output of the space-time decoder. A spatial Fourier (virtual) representation of the channel is exploited to allow for the analysis of MIMO channels with quite general fading statistics. Conditions, known as generalized orthogonality conditions (GOC), are derived for an LD code to achieve an upper bound on the mutual information, with the understanding that LD codes that achieve the upper bound, if they exist, are optimal. Explicit code constructions and properties of the optimal power allocation schemes are also derived. In particular, it is shown that optimal LD codes correspond to beamforming to a single virtual transmit angle at low SNR, and a necessary and sufficient condition for beamforming to be optimal is provided. Finally, numerical results are provided to illustrate the optimal code design for two examples of sparse scattering environments. The performance of the optimal LD codes for these scattering environments is compared with that of LD codes designed assuming the i.i.d. Rayleigh (rich scattering) model, and it is shown that the optimal LD codes perform significantly better. The optimal LD codes are also compared to beamforming LD codes and it is shown that beamforming is nearly optimal over a range of SNR's of interest.

In this paper, two-sender unicast index coding problem (TUICP) is studied, where the senders possibly have some messages in common, and each receiver requests a unique message. It is analyzed using three independent sub-problems (which are single-sender unicast index coding problems (SUICPs)) and the interactions among them. These sub-problems are described by three disjoint vertex-induced subgraphs of the side-information graph of the TUICP respectively, based on the availability of messages at the senders. The TUICP is classified based on the type of interactions among the sub-problems. Optimal scalar linear index codes for a class of TUICP are obtained using those of the sub-problems. For two classes, we identify a sub-class for which scalar linear codes are obtained using the notion of joint extensions of SUICPs. An SUICP ${{\mathcal{I}}_E}$ is said to be a joint extension of l SUICPs if the fitting matrices of all the l SUICPs are disjoint submatrices of that of ${{\mathcal{I}}_E}$. Joint extensions generalize the notion of rank-invariant extensions. Scalar linear codes and a condition for optimality of the codes are given for a class of joint extensions. Using this result, scalar linear codes and the conditions for their optimality are obtained for two classes of the TUICP.

This paper studies linear and affine error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to having an information rate at most (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial linear codes correcting deletions exist, i.e., that the -fold repetition codes and its rate of are basically optimal for any . We disprove this and show the existence of binary linear codes of length and rate just below capable of correcting insertions and deletions. This identifies rate as a sharp threshold for recovery from deletions for linear codes and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically good linear code for Hamming errors into an asymptotically good linear code for insdel errors. Last, we show that the -rate limitation does not hold for affine codes by giving an explicit affine code of rate which can efficiently correct a constant fraction of insdel errors.