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Abstract: Let be the set of symmetric bilinear forms on an m-dimensional vector space over , where q is a power of two. A subset Y of is called an -set if the difference of every two distinct elements in Y has rank at least d. Such objects are closely related to certain families of codes over Galois rings of characteristic four. An upper bound on the size of -sets is derived, and in certain cases, the rank distance distribution of an -set is explicitly given. Constructions of -sets are provided for all possible values of m and d. [ABSTRACT FROM AUTHOR]

A constant-amplitude code is a code that reduces the peak-to-average power ratio (PAPR) in multicode code-division multiple access (MC-CDMA) systems to the favorable value 1. In this paper, quaternary constant-amplitude codes (codes over Z4) of length 2m with error-correction capabilities are studied. These codes exist for every positive integer m, while binary constant-amplitude codes cannot exist if m is odd. Every word of such a code corresponds to a function from the binary m-tuples to Z4 having the bent property, i.e., its Fourier transform has magnitudes 2m/2. Several constructions of such functions are presented, which are exploited in connection with algebraic codes over Z4 (in particular quaternary Reed-Muller, Kerdock, and Delsarte-Goethals codes) to construct families of quaternary constant-amplitude codes. Mappings from binary to quaternary constant-amplitude codes are presented as well. [ABSTRACT FROM AUTHOR]

The use of error-correcting codes for tight control of the peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency-division multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each q-phase (q is even) sequence of length 2m lies in a complementary set of size 2k+1, where k is a nonnegative integer that can be easily determined from the generalized Boolean function associated with the sequence. For small k this result provides a reasonably tight bound for the PMEPR of q-phase sequences of length 2m. A new 2 h-ary generalization of the classical Reed-Muller code is then used together with the result on complementary sets to derive flexible OFDM coding schemes with low PMEPR. These codes include the codes developed by Davis and Jedwab as a special case. In certain situations the codes in the present correspondence are similar to Paterson's code constructions and often outperform them. [ABSTRACT FROM AUTHOR]

The use of error-correcting codes for tight control of the peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency-division multiplexing (OFDM) transmission is considered. A well-known approach for the construction of such codes is to take a code that is good in the classical coding-theoretic sense and to choose a translate of this code that minimizes the PMEPR. A fundamental problem is to determine the minimum PMEPR over all translates of a given code. Motivated by a recent lower bound for this minimum, an existence result is presented here. Roughly speaking, given a code C of sufficiently large length n, there exists a translate of C with PMEPR at most k\log(|C|n^1+ε) for all ε>0 and for some k independent of n. This result is then used to show that for n ≥ 32 there is a translate of the lengthened dual of a binary primitive t-error-correcting BCH code with PMEPR at most 8(t+2)\log n. [ABSTRACT FROM PUBLISHER]