Abstract Combinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for $$t > 1$$ t > 1 were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher’s inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength $$t=2$$ t = 2 , resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over $$\mathbb {F}_2$$ F 2 and $$\mathbb {F}_3$$ F 3 .

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.

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 √2nlog(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 √n log log n and, therefore, grows strictly more slowly than the peak sidelobe level of a typical binary sequence.

For integerrsatisfying 0 ≤r≤p-2, a sequence family Ωrof polyphase sequences of prime periodp, size (p-2)pr, and maximum correlation at most 2 +(r+1) √(p)is presented. The sequence families are nested, that is, Ωris contained in Ωr+ 1, which provides design flexibility with respect to family size and maximum correlation. The sequences in Ωrare 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 Ω0, which was previously proposed by Scholtz and Welch. Sequence family Ω2is 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.

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 value1. In this paper, quaternary constant-amplitude codes (codes overZ4) of length2mwith error-correction capabilities are studied. These codes exist for every positive integerm, while binary constant-amplitude codes cannot exist ifmis odd. Every word of such a code corresponds to a function from the binarym-tuples to Z4having the bent property, i.e., its Fourier transform has magnitudes2m/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.

In this paper, Zopf4-valued quadratic forms defined on a vector space overGF(2)are studied. A classification of such forms is established, distinguishing Zopf4-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. Whent=0 ormis odd, the correlation distribution of familyS(t), consisting of quaternary sequences of length2m-1, is established. Then, motivated by practical considerations, a subsetS*(t) of familyS(t) is defined, and the correlation distribution of familyS*(t) is given for odd and evenm.

Zopf4-valued quadratic forms associated with symmetric bilinear forms are studied. A classification of such forms according to their type and rank is derived. This result is used to compute the distribution of certain exponential sums over Galois rings, which occur frequently in the analysis of correlation properties of quaternary sequence sets. The framework is then illustrated by determining the possible correlation values of family S(t) of length 2m- 1 proposed by Kumar et al.. For odd m the correlation distribution is derived, which involves the computation of the rank distribution of certain symmetric codes in the rank metric.

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 2h-ary phase-shift keying (2h-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.

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 Zopf4) of length 2mwith 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 Zopf4having 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 Zopf4(in particular quaternary Reed-Muller and trace codes) to construct families of quaternary constant-amplitude codes. Mappings from binary to quaternary constant-amplitude codes are presented as well. The resulting coding options allow a rich trade-off between code rate and minimum distance.

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 2mlies 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 2h-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

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 2morganizes in m!/2 cosets of a generalized first-order Reed-Muller code RMq(1, m). In this paper a more general construction technique for cosets of RMq(1, m) with low PMEPR is provided. These cosets contain so-called near-complementary sequences. The application of this result is then illustrated by providing some construction examples. First, it is shown that the m!/2 cosets of RMq(1, m) comprised of Golay sequences just arise as a special case. Second, further families of cosets of RMq(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 RMq(1, m) is proved as well, and it is demonstrated that the upper bound on the PMEPR is tight in many cases

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 2h-ary phase-shift keying (PSK) constellation? A lower bound on the PMEPR is established, which is related to the covering radius of the binary code. Most notably, a small covering radius of the binary code precludes a significant PMEPR reduction. The bound is then applied to some well understood codes, including BCH codes and their duals, Reed-Muller codes, and convolutional codes. It appears that some previously obtained (presumably not optimal) results attain or approach our bound

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).

In this paper new codes for orthogonal frequency-division multiplexing (OFDM) with tightly controlled peak-to-mean envelope power ratio (PMEPR) are proposed. We identify a new family of sequences occurring in complementary sets and show that such sequences form subsets of a new generalization of the Reed-Muller codes. Contrarily to previous constructions we present a compact description of such codes, which makes them suitable even for larger block lengths. We also show that some previous constructions just occur as special cases in our construction