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We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled. [ABSTRACT FROM AUTHOR]

A subset Y of the general linear group $\text{GL}(n,q)$ is called t -intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t -dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$. We show that, if n is sufficiently large compared to t , the size of every such t -intersecting set is at most that of the stabiliser of a basis of a t -dimensional subspace of $\mathbb{F}_q^n$. In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t -dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$. These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q -analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$. [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 [Z.sub.4]) of length [2.sup.m] 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 [Z.sup.4] having the bent property, i.e., its Fourier transform has magnitudes [2.sup.m/2]. Several constructions of such functions are presented, which are exploited in connection with algebraic codes over [Z.sub.4] (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. Index Terms--Bent function, code, code-division multiple access (CDMA), Delsarte-Goethals, Kerdock, multicode, peak-to-average power ratio (PAPR), quaternary, Reed-Muller.

The peak-to-mean envelope power ratio (PMEPR) of a code employed in orthogonal frequency-division multiplexing (OFDM) systems can be reduced by permuting its coordinates and by rotating each coordinate by a fixed phase shift. Motivated by some previous designs of phase shifts using suboptimal methods, the following question is considered in this paper. For a given binary code, how much PMEPR reduction can be achieved when the phase shifts are taken from a [2.sup.h]-ary phase-shift keying ([2.sup.h]-PSK) constellation? A lower bound on the achievable PMEPR is established, which is related to the covering radius of the binary code. Generally speaking, the achievable region of the PMEPR shrinks as the covering radius of the binary code decreases. The bound is then applied to some well understood codes, including nonredundant BPSK signaling, BCH codes and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated that most (presumably not optimal) phase-shift designs from the literature attain or approach our bound. Index Terms--BCH codes, convolutional codes, covering radius, orthogonal frequency-division multiplexing (OFDM), peak-to-average power ratio (PAPR), peak-to-mean envelope power ratio (PMEPR), Reed-Muller codes.

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 [2.sup.m] lies in a complementary set of size [2.sup.k+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 [2.sup.m]. A new [2.sup.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. Index Terms--Code, complementary, correlation, Golay, orthogonal frequency-division multiplexing (OFDM), peak-to-mean envelope power ratio (PMEPR), Reed--Muller, sequence, set.

Golay sequences are well suited for use as codewords in orthogonal frequency-division multiplexing (OFDM), since their peak-to-mean envelope power ratio (PMEPR) in q-ary phase-shift keying (PSK) modulation is at most 2. It is known that a family of polyphase Golay sequences of length [2.sup.m] organizes in m!/2 cosets of a q-ary generalization of the first-order Reed--Muller code, [RM.sub.q] (1, m). In this paper, a more general construction technique for cosets of [RM.sub.q] (1, m) with low PMEPR is established. These cosets contain so-called near-complementary sequences. The application of this theory is then illustrated by providing some construction examples. First, it is shown that the m!/2 cosets of [RM.sub.q] (1, m) comprised of Golay sequences just arise as a special case. Second, further families of cosets of [RM.sub.q] (1, m) with maximum PMEPR between 2 and 4 are presented, showing that some previously unexplained phenomena can now be understood within a unified framework. A lower bound on the PMEPR of cosets of [RM.sub.q] (1, m) is proved as well, and it is demonstrated that the upper bound on the PMEPR is tight in many cases. Finally, it is shown that all upper bounds on the PMEPR of cosets of [RM.sub.q] (1, m) also hold for the peak-to-average power ratio (PAPR) under the Walsh--Hadamard transform (WHT). Index Terms--Aperiodic, code, complementary, correlation, Golay, orthogonal frequency-division multiplexing (OFDM), peak-to-average power ratio (PAPR), peak-to-mean envelope power ratio (PMEPR), Reed--Muller, Rudin--Shapiro, sequence, Walsh--Hadamard.

The Pursley-Sarwate criterion of a pair of finite complex-valued sequences measures the collective smallness of the aperiodic autocorrelations and the aperiodic cross-correlations of the two sequences. It is known that this quantity is always at least 1 with equality if and only if the sequence pair is a Golay pair. We exhibit pairs of complex-valued sequences whose entries have unit magnitude for which the Pursley-Sarwate criterion tends to 1 as the sequence length tends to infinity. Our constructions use different carefully chosen Chu sequences. [ABSTRACT FROM AUTHOR]

A 3-phase Barker array is a matrix of third roots of unity for which all out-of-phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two-dimensional 3-phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3-phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double-exponentially growing arithmetic function T such that no 3-phase Barker array of size [ABSTRACT FROM AUTHOR]

Let $A_n=(a_0,a_1,\ldots ,a_{n-1})$ be drawn uniformly at random from ${\{ }-1,+1{\} }^n$ and define\[M(A_n)=\max _{01}{\mathrm {}}.\]It is proved that $M(A_n)/\sqrt {n\log n}$ converges in probability to $\sqrt {2}$. This settles a problem first studied by Moon and Moser in the 1960s and proves in the affirmative a recent conjecture due to Alon, Litsyn, and Shpunt. It is also shown that the expectation of $M(A_n)/\sqrt {n\log n}$ tends to $\sqrt {2}$. [ABSTRACT FROM AUTHOR]

A binary sequence of length n is an n-tuple with elements in \-1,1\ and its peak sidelobe level is the largest absolute value of its aperiodic autocorrelations at nonzero shifts. A classical problem is to find binary sequences whose peak sidelobe level is small compared to the length of the sequence. Using known techniques from probabilistic combinatorics, this paper gives a construction for a binary sequence of length n with peak sidelobe level at most \sqrt2n\log (2n) for every n > 1. This improves the best known bound for the peak sidelobe level of a family of explicitly constructed binary sequences, which arises for the family of m-sequences. By numerical analysis, it is argued that the peak sidelobe level of the constructed sequences grows in fact like order \sqrtn\log\log n and, therefore, grows strictly more slowly than the peak sidelobe level of a typical binary sequence. [ABSTRACT FROM AUTHOR]

For integer r satisfying 0\leq r\leq p-2, a sequence family \Omega_r of polyphase sequences of prime period p, size (p-2)p^r, and maximum correlation at most 2+(r+1)\sqrtp is presented. The sequence families are nested, that is, \Omega_r is contained in \Omegar+1, which provides design flexibility with respect to family size and maximum correlation. The sequences in \Omega_r are derived from a combination of multiplicative and additive characters of a prime field. Estimates on hybrid character sums are then used to bound the maximum correlation. This construction generalizes \Omega_0, which was previously proposed by Scholtz and Welch. Sequence family \Omega_2 is closely related to a recent design by Wang and Gong, who bounded its maximum correlation using methods from representation theory and asked for a more direct proof of this bound. Such a proof is given here and an improvement of the bound is provided. [ABSTRACT FROM AUTHOR]

In this paper, 74-valued quadratic forms defined on a vector space over GF(2) are studied. A classification of such forms is established, distinguishing Z4-valued quadratic forms only by their rank and whether the associated bilinear form is alternating. This result is used to compute the distribution of certain exponential sums, which occur frequently in the analysis of quaternary codes and quaternary sequence sets. The concept is applied as follows. When t = 0 or m is odd, the correlation distribution of family S (t), consisting of quaternary sequences of length 2m - 1, is established. Then, motivated by practical considerations, a subset S* (t) of family S (t) is defined, and the correlation distribution of family S* (t) is given for odd and even m. [ABSTRACT FROM AUTHOR]

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from F q n to F q to the set of affine functions from F q n to F q. We prove the conjecture for each q such that the characteristic of F q lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the [ q n , n + 1 ] Reed-Muller code over F q and so answers a question raised by Leducq in 2013. Our results extend the case q = 2 , which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory. [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]