Stephanie Perkins

Binary Huffman equivalent codes with a short synchronizing codeword

For a given set of codeword lengths, there are many different optimal variable-length codes, which are all Huffman (1952) equivalent codes. Some of these codes may contain a synchronizing codeword which resynchronizes the code whenever it is transmitted. The shorter the synchronizing codeword, the quicker the code will resynchronize. Ferguson and Rabinowitz (1984) suggest the problem of finding, for a given set of codeword lengths, the binary Huffman equivalent code with the shortest synchronizing codeword. We consider binary Huffman equivalent codes whose shortest codeword has length m>1 and which contain a synchronizing codeword of length m+1, the shortest possible in this case. We provide an algorithm for constructing these codes for a given set of codeword lengths, if such a code exists. We study further properties of these codes and show that when in m/spl ges/3 the codes contain more than one synchronizing codeword. Finally, we suggest ways of improving the synchronization properties of the codes and provide some example codes.

Linear and nonlinear constructions of DNA codes with Hamming distance d, constant GC-content and a reverse-complement constraint

In a previous paper, the authors used cyclic and extended cyclic constructions to obtain codes over an alphabet {A,C,G,T} satisfying a Hamming distance constraint and a GC-content constraint. These codes are applicable to the design of synthetic DNA strands used in DNA microarrays, as DNA tags in chemical libraries and in DNA computing. The GC-content constraint specifies that a fixed number of positions are G or C in each codeword, which ensures uniform melting temperatures. The Hamming distance constraint is a step towards avoiding unwanted hybridizations. This approach extended the pioneering work of Gaborit and King. In the current paper, another constraint known as a reverse-complement constraint is added to further prevent unwanted hybridizations. Many new best codes are obtained, and are reproducible from the information presented here. The reverse-complement constraint is handled by searching for an involution with 0 or 1 fixed points, as first done by Gaborit and King. Linear codes and additive codes over GF(4) and their cosets are considered, as well as shortenings of these codes. In the additive case, codes obtained from two different mappings from GF(4) to {A,C,G,T} are considered.

Linear and nonlinear constructions of DNA codes with Hamming distance d and constant GC-content

Coding theory has several applications in genetics and bioengineering. This paper constructs codes over an alphabet {A,C,G,T} relevant to the design of synthetic DNA strands used in DNA microarrays, as DNA tags in chemical libraries and in DNA computing. The codes are designed to avoid unwanted hybridizations and to ensure uniform melting temperatures. Specifically, the codes considered here satisfy a Hamming distance constraint and a GC-content constraint. In comparison with previous work, longer codes are constructed, the examination of cyclic and extended cyclic codes is more comprehensive, attention is paid to the mapping from field or ring elements to {A,C,G,T}, cosets of codes are used and a nonlinear shortening operation is performed. Many new best codes are constructed, and are reproducible from the information presented here.

Construction of Heuristics for a Search-Based Approach to Solving Sudoku

Sudoku is a logic puzzle, consisting of a 9×9 grid and further subdivided into ‘mini-grids’ of size 3×3. Each row, column, and 3×3 mini-grid contains the numbers 1 to 9 once, with a true Sudoku grid having a unique solution. Sudoku, along with similar combinatorial structures, has relationships with a range of real-world problems. Much published work on the solution of Sudoku puzzles has acknowledged the link between Sudoku and Latin Squares, thereby recognising the scale of any search space of possible solutions and that the generalization of the puzzle to larger grid sizes is NPcomplete. However, most published approaches to the solution of Sudoku puzzles have focussed on the use of constraint satisfaction algorithms that effectively mimic solution by hand, rather than directly exploiting features of the problem domain to reduce the size of the search space and constructing appropriate heuristics for the application of search techniques. This paper highlights important features of the search space to arrive at heuristics employed in a modified steepest ascent hill-climbing algorithm, and proposes a problem initialization and neighbourhood that greatly speed solution through a reduction of problem search space. Results shown demonstrate that this approach is sufficient to solve even the most complex rated puzzles, requiring relatively few moves. An analysis of the nature of the problem search space is offered.

Exploiting Spatial Separations in CDMA Systems With Correlation Constrained Sets of Hadamard Matrices

Spreading codes for code-division multiple-access (CDMA) systems are constructed based on Hadamard matrices and almost-bent functions. These codes can be assigned to a tessellation of hexagonal cells in such a way that codewords assigned to the same cell or to adjacent cells have zero cross-correlation. The nonzero cross-correlations in the code only apply to pairs of codewords assigned to nonadjacent cells. The codes have four times as many codewords as codes constructed from a single Hadamard matrix, which leads to significantly increased codeword re-use distances in comparison with the conventional use of a single Hadamard matrix with the same number of codewords per cell.

Robust Data Compression: Consistency Checking in the Synchronization of Variable Length Codes

Many data compression techniques are essentially variable length codes. Such codes suffer from a vulnerability to error propagation. This occurs when an error causes the correspondence between the data stream and the stream of encoded symbols to be lost and continues until synchronization between the two is re-established. This paper compares a number of strategies for synchronization in which redundant synchronizing words are used with a mechanism for labelling or counting them. In each case the strategy is investigated for the prevention of error propagation, ensuring the re-establishment of the correspondence with the correct point in the data stream. Three classes of potential synchronization error are explored: the false recognition of spurious words, the loss of synchronizing words and synchronization misinterpretation, when one synchronizing word is confused with another. The probabilities of synchronization error are analysed. It is shown that the strategies are successful against random errors when they incorporate some mechanism for consistency checking. The relative merits of different schemes are demonstrated. The schemes are also compared with the technique known as bitstuffing. This is considered on its own and in combination with the other schemes.

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.

Automation of the Solution of Kakuro Puzzles

Kakuro puzzles, also called cross sum puzzles, are grids containing clues to the completion of numerical ‘words’. Being structured in a similar way to crossword puzzles, Kakuro grids contain overlapping continuous runs that are exclusively either horizontal or vertical. The ‘clues’ take the form of specified run totals, and a puzzle is solved by placing a value in each cell such that every run sums to its specified total, and no run contains duplicate values. While most puzzles have only a single solution, longer runs may be satisfied using many arrangements of values, leading to the puzzle having a deceptively large search space. The associated, popular Sudoku puzzle has been linked with important real-world applications including timetabling and conflict free wavelength routing, and more recently, coding theory due to its potential usefulness in the construction of erasure correction codes. It is possible that Kakuro puzzles will have similar applications, particularly in the construction of codes, where run totals may form a generalised type of parity check. A project has begun to investigate the properties of the class of Kakuro puzzles, and thereby establish its potential usefulness to real-world applications including coding theory. This paper reports some early findings from that project, specifically concerning puzzle complexity and the appropriateness of heuristic approaches for its automated solution. It highlights the use of heuristics to guide search by a backtracking solver, in preference to local search optimisation, and reports on the effectiveness of two heuristics and a pruning technique for reducing solution time. The authors believe this to be the first published work in the use of heuristics, in combination with pruning, for the automated solution of Kakuro puzzles.

Extended synchronizing codewords for q -ary complete prefix codes

Abstract Variable length codes are often used for data transmission. However, a single bit error may cause the loss of some or all subsequent data due to loss of decoder synchronization. It is necessary to devise schemes that resynchronize the decoder to allow the data to again be decoded correctly. Some applications require that the scheme also maintains the position of the decoded symbols in the data stream. Recently, Lam and Kulkarni introduced the concept of an extended synchronizing codeword (ESC) which can be used with an appropriate scheme to prevent propagation errors and also symbol shift errors. We extend their work and show how ESCs can be generated with minimal overhead.

A scheme for the synchronization of variable length codes

Abstract A synchronization scheme is necessary when variable length codes are used in the presence of errors. In this paper we present a scheme for synchronizing the data stream without allowing slippage. We achieve this by inserting a number of distinct keywords, each consisting of a synchronizing sequence and an explicit or implicit cyclic count, into the data at intervals. We present decoding algorithms for this scheme and prove their effectiveness given limits on the maximum number of errors per cycle of keywords.