Special Codewords Research Articles

Ensemble of high performance structured binary convolutional LDPC codes with moderate rates

An algebraic construction methodology is proposed to design binary time-invariant convolutional low-density parity-check (LDPC) codes. Assisted by a proposed partial search algorithm, the polynomialform parity-check matrix of the time-invariant convolutional LDPC code is derived by combining some special codewords of an (n, 2, n − 1) code. The achieved convolutional LDPC codes possess the characteristics of comparatively large girth and given syndrome former memory. The objective of our design is to enable the time-invariant convolutional LDPC codes the advantages of excellent error performance and fast encoding. In particular, the error performance of the proposed convolutional LDPC code with small constraint length is superior to most existing convolutional LDPC codes.

Error-correction of linear codes via colon ideals

We show that errors in data transmitted through linear codes can be thought of as codewords of minimum weight of new linear codes. To determine errors we can then use methods specific to finding such special codewords. One of these methods consists of finding the primary decomposition of the saturation of a certain homogeneous ideal. When good words (i.e. vectors with a unique nearest neighbor) are error-corrected, the saturated ideal is just the prime ideal of a point (so the primary decomposition is superfluously determined); we show that this ideal can be computed by coloning the original homogeneous ideal with a power of a certain variable. We then determine the smallest such power for any linear code.

A Turbo Coding Scheme for Channels with Synchronization Errors

We describe a turbo coding scheme with nonsystematic recursive convolutional codes for channels that exhibit insertions and deletions along with substitution errors. The scheme uses the MAP decoding algorithm for this channel and a special codeword arrangement to simplify the synchronization process. The decoder uses a special trellis structure to extract synchronization extrinsic information that is exchanged between the two constituent decoders. The proposed scheme is shown to provide improvement over earlier channel coding approaches for this channel model.

Error-resilient pattern classification using a combination of spreading and coding gains

An approach that aims to enhance error resilience in pattern classification problems is proposed. The new approach combines the spread spectrum technique, specifically its selectivity and sensitivity, with error-correcting output codes (ECOC) for pattern classification. This approach combines both the coding gain of ECOC and the spreading gain of the spread spectrum technique to improve error resilience. ECOC is a well-established technique for general purpose pattern classification, which reduces the multi-class learning problem to an ensemble of two-class problems and uses special codewords to improve the error resilience of pattern classification. The direct sequence code division multiple access (DS-CDMA) technique is a spread spectrum technique that provides high user selectivity and high signal detection sensitivity, resulting in a reliable connection through a noisy radio communication channel shared by multiple users. Using DS-CDMA to spread the codeword, assigned to each pattern class by the ECOC technique, gives codes with coding properties that enable better correction of classification errors than ECOC alone. Results of performance assessment experiments show that the use of DS-CDMA alongside ECOC boosts error-resilience significantly, by yielding better classification accuracy than ECOC by itself.

On Components of Preparata Codes

The paper considers the interrelation between i-components of an arbitrary Preparata-like code P and i-components of a perfect code C containing P. It is shown that each i-component of P can uniquely be completed to an i-component of C by adding a certain number of special codewords of C. It is shown that the set of vertices of P in a characteristic graph of an arbitrary i-component of C forms a perfect code with distance 3.

On some cosets of the first-order Reed-Muller code with high minimum weight

We study a family of particular cosets of the first-order Reed-Muller code R(1,m): those generated by special codewords, the idempotents. Thus we obtain new maximal weight distributions of cosets of R(1,7) and 84 distinct almost maximal weight distributions of cosets of R(1,9), that is, with minimum weight 240. This leads to crypotographic applications in the context of stream ciphers.

Synchronizing codewords of q-ary Huffman codes

Abstract Some Huffman codes contain a special codeword called a synchronizing codeword, which resynchronizes the code whenever it is transmitted. We demonstrate properties of synchronizing codewords and, in particular, give some relationships between the length of the shortest codeword and the length and structure of the synchronizing codeword. A tight upper bound and some lower bounds are presented. We show that given a shortest codeword of length m and a synchronizing codeword of length r m − 1 then the code also contains other synchronizing codewords. We calculate the number and lengths of these codewords. Finally, several examples of good codes are given.

Synchronizing codewords of q-ary Huffman codes

Some Huffman codes contain a special codeword called a synchronizing codeword, which resynchronizes the code whenever it is transmitted. We demonstrate properties of synchronizing codewords and, in particular, give some relationships between the length of the shortest codeword and the length and structure of the synchronizing codeword. A tight upper bound and some lower bounds are presented. We show that given a shortest codeword of length m and a synchronizing codeword of length r<2 m − 1 then the code also contains other synchronizing codewords. We calculate the number and lengths of these codewords. Finally, several examples of good codes are given.

An extended run-length encoder and decoder for compression of black/white images

Most of the data compression strategies currently used for black/white (one-bit gray-scale) images involve the detection and noiseless encoding of a specified set of image features, the most common features being "runs" of adjacent equal.valued picture elements. A procedure is provided that, given the frequency distribution for a particular set of features in a sample of images, calculates an extended run-length code (ERLC) that includes two classes of codewords: "regular" codewords for the runs and "special" codewords for the other image features. An ERLC is shown to be highly efficient for the sample images, uniformly efficient for similar images not in the sample, and easily implemented. An encoder and the corresponding decoder are described which can implement any ERLC that includes up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> regular codewords and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> special codewords, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> are design parameters. Neither the encoder nor the decoder requires a table of codewords; rather, both are controlled by two auxiliary tables that include only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\ell_{max} [\log_{2} R]</tex> -bit entries and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\ell_{max} [\log_{2} S]</tex> -bit entries, respectively, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\ell_{max}</tex> is the length of the longest codeword.

Transmitting Television as Clusters of Frame-to-Frame Differences

A number of redundancy reduction techniques are used in a coder that is about eight times more efficient than simple PCM. The coder is capable of transmitting Picturephone® signals at an average rate of one bit per picture-element (2 Megabits per second). When there is movement in the scene, most transmission time is devoted to the parts of the picture that change significantly. The data are generated irregularly but the data flow is smoothed prior to transmission in a buffer that holds about one frame of data. The redundancy reduction techniques used and the behavior of the coder are discussed both from an intuitive and from a statistical viewpoint. The positions of elements that change are signaled by addressing the first element of a run of changes and marking the end of the run with a special code word. The changes of luminance are transmitted as frame-to-frame differences using variable-length code words. When rapid motion makes the buffer more than a quarter full, only differences for every second element are transmitted, the values of the intervening changed elements being set equal to the average of their neighbors. If the buffer continues to fill, the threshold that determines which changes are significant is raised from 4/256 to 7/256 of the maximum signal value. When violent motion causes the buffer to fill completely, replenishment is stopped for about one frame while the buffer empties. Subsampling and raising the threshold are not objectionable because viewers rarely detect the small impairments introduced in moving images. Observers are critical, however, of small impairments in stationary scenes. Thus, to maintain high quality in stationary areas, the entire picture is forcibly updated every three seconds by transmitting 8-bit luminance values for three lines of every frame. A record of the coder's behaviour is available as a 16-milimeter movie film.