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.

Variable length codes offer advantages for data compression, but are susceptible to loss of synchronization if a bit error occurs. This can be mitigated by the use of variable length codes with specific mechanisms for resynchronization. Such mechanisms exist in the codes known as Huffman Equivalent (HE) codes and T-codes. These have been extensively studied in the literature. For optimal compression a length vector is obtained from the probabilities of the symbols of the alphabet used. However, HE-codes and T-codes do not exist for all length vectors. In this paper, a new class of variable length codes with good synchronization properties is developed. These are referred to as ordered termination (OT) codes. OT-codes do exist for all length vectors. Experimental results and some theoretical support suggest that OT-codes compare favorably with HE- and T-codes.

A mixed code is an error-correcting code in which different entries of the codewords can be chosen from different alphabets. In this correspondence an upper bound is given for the number of codewords in a mixed code where all the entries can come from distinct alphabets. This bound improves the sphere packing bound in several directions. The result is specialized to a simpler form in the case when only two distinct alphabets are used. Numerical results are presented to show that, in various cases, two different forms of the bound and alternative choices of a parameter may give the strongest bound.

When variable length codes are used in the presence of error, slippage can occur. The number of symbols decoded before synchronization is regained can differ from the number of encoded symbols, causing misinterpretation of data. In previous work certain synchronization schemes have been shown to be satisfactory in preventing permanent loss of synchronization and slippage in the presence of random error. In this paper we demonstrate that these schemes can also be satisfactory in the presence of burst errors, and that their relative merits remain the same.

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.

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.