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In this paper we use group theoretic tools to obtain random variables which violate linear rank inequalities, that is inequalities which always hold on ranks of subspaces. We consider ten of the 24 (non-Shannon type) generators of linear rank inequalities in five variables and look at them as group inequalities. We prove that for primes $p,q$, groups of order $pq$ always satisfy these ten group inequalities. We give partial results for groups of order $p^2q$, and find that the symmetric group $S_4$ is the smallest group to yield violations for two among the ten group inequalities.

We study fast lattice decodability of space-time block codes for $n$ transmit and receive antennas, written very generally as a linear combination $\sum_{i=1}^{2l} s_i A_i$, where the $s_i$ are real information symbols and the $A_i$ are $n\times n$ $\mathbb R$-linearly independent complex valued matrices. We show that the mutual orthogonality condition $A_iA_j^* + A_jA_i^*=0$ for distinct basis matrices is not only sufficient but also necessary for fast decodability. We build on this to show that for full-rate ($l = n^2$) transmission, the decoding complexity can be no better than $|S|^{n^2+1}$, where $|S|$ is the size of the effective real signal constellation. We also show that for full-rate transmission, $g$-group decodability, as defined in [1], is impossible for any $g \ge 2$. We then use the theory of Azumaya algebras to derive bounds on the maximum number of groups into which the basis matrices can be partitioned so that the matrices in different groups are mutually orthogonal---a key measure of fast decodability. We show that in general, this maximum number is of the order of only the $2$-adic value of $n$. In the case where the matrices $A_i$ arise from a division algebra, which is most desirable for diversity, we show that the maximum number of groups is only $4$. As a result, the decoding complexity for this case is no better than $|S|^{\lceil l/2 \rceil}$ for any rate $l$., submitted to IEEE Transactions on Information Theory

We propose a full-rate iterated space-time code construction, to design 2n-dimensional codes from n-dimensional cyclic algebra based codes. We give a condition to determine whether the resulting codes satisfy the full-diversity property, and study their maximum likelihood decoding complexity with respect to sphere decoding. Particular emphasis is given to the cases n = 2, sometimes referred to as MIDO (multiple input double output) codes, and n = 3. In the process, we derive an interesting way of obtaining division algebras, and study their center and maximal subfield., 32 pages, 2 figures

Let G be a finite nilpotent group and K a number field with torsion relatively prime to the order of G. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of K ramified in a Galois extension of K with Galois group isomorphic to G. This sharpens and extends results of Geyer and Jarden and of Plans. Alternatively, we show how to use Frohlich’s result on realizing the Schur multiplicator in order to realize a family of groups given by central extensions with minimal ramification.

In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant −19. We prove that any such curve can be given by an explicit equation of certain form (see Theorem 5.1). Using these equations we obtain a table of maximal and minimal curves over prime finite fields with discriminant −19 of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.

The authors give a construction of multiblock space-time block codes from cyclic division algebras with a given nth root of unity as the nonnorm element. The construction uses local class field theory. Multiblock codes with M blocks can achieve an M-fold increase in diversity.

We introduce two new infinite families of APN functions, one on fields of order 22k for k not divisible by 2, and the other on fields of order 23k for k not divisible by 3. The polynomials in the first family have between three and k+2 terms, the second family's polynomials have three terms.

We study fast decodability for general rate spacetime block codes. We derive bounds on the number of generators that can be in mutually orthogonal groups and use these to show that that the ML decoding complexity of a full-rate n×n spacetime block code is unfortunately quite high: at least of the order of ∣S∣n2+1, where S is the effective real constellation. We also show that g-group decodability is not possible for full-rate codes.

We present an iterative construction of algebraic space-time codes. Starting from a division algebra D, we show how to embed it into a larger ring A = A(D) and give conditions for A to be a new division algebra. Starting from a quaternion division algebra D 1 , we thus obtain a sequence D 1 ⊂ D 2 ⊂… of division algebras where D i = A(D i−1 ). Each of the D i can be used as an underlying structure to build a space-time C i . Furthermore, the iteration step is done such that fast-decodability of the original code is preserved. We illustrate our technique by creating an iterative version of the Silver code.

In this paper, fast-decodable lattice code constructions are designed for the nonorthogonal amplify-and-forward (NAF) multiple-input multiple-output (MIMO) channel. The constructions are based on different types of algebraic structures, e.g. quaternion division algebras. When satisfying certain properties, these algebras provide us with codes whose structure naturally reduces the decoding complexity. The complexity can be further reduced by shortening the block length, i.e., by considering rectangular codes called less than minimum delay (LMD) codes., 5 pages

A set of quasi-uniform random variables $X_1,...,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in more detail different families of non-abelian groups with respect to the entropic vectors they yield. In particular, we address the question of whether a given non-abelian group $G$ and some fixed subgroups $G_1,...,G_n$ end up giving the same entropic vector as some abelian group $A$ with subgroups $A_1,...,A_n$, in which case we say that $(A, A_1,..., A_n)$ represents $(G, G_1, ..., G_n)$. If for any choice of subgroups $G_1,...,G_n$, there exists some abelian group $A$ which represents $G$, we refer to $G$ as being abelian (group) representable for $n$. We completely characterize dihedral, quasi-dihedral and dicyclic groups with respect to their abelian representability, as well as the case when $n=2$, for which we show a group is abelian representable if and only if it is nilpotent. This problem is motivated by understanding non-linear coding strategies for network coding, and network information theory capacity regions., Comment: 14 pages

We propose an iterative full-rate code construction for multiple antenna channels, with 6Tx and 3Rx antennas. We start with two 3x 3 codes arising from a cyclic division algebra, and combine them to obtain a 6 x6 codeword. We provide a sufficient criterion for when the resulting code is fully diverse. Furthermore, this construction preserves fast-decodability of the original code. Accepted version

We consider the problem of designing codes for multiple antenna channels, with 4 transmit antennas and two receive antennas, using as code design full diversity and fast decodability. We present an iterated construction, which allows us to start with any algebraic 2 × 2 space-time code, and map it to a 4 × 4 MIDO code, for which a criterion for full diversity is available. We illustrate our method with two MIDO codes, built upon the Silver Code and the Alamouti code, for which the decoding complexity is analyzed. Simulation results show that the proposed codes exhibit very good behavior compared to existing codes. Accepted version

In the context of space-time block codes (STBCs), the theory of generalized quaternion and biquaternion algebras (i.e., tensor products of two quaternion algebras) over arbitrary base fields is presented, as well as quadratic form theoretic criteria to check if such algebras are division algebras. For base fields relevant to STBCs, these criteria are exploited, via Springer's theorem, to construct several explicit infinite families of (bi-)quaternion division algebras. These are used to obtain new $2\x 2$ and $4\x 4$ STBCs., 8 pages, final version

We present two infinite families of APN functions where the degree of the field is divisible by 3 but not 9. Our families contain two already known families as special cases. We also discuss the inequivalence proof (by computation) which shows that these functions are new., 12 pages, submitted to Cryptography and Communications

In this paper we compute the Fourier spectra of some recently discovered binomial APN functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting BCH code. Furthermore, for fields of odd degree, our results provide an alternative proof of the APN property of the functions., 20 pages. Submitted to the SIAM Journal on Discrete Mathematics

We compute the Walsh spectrum of a new quadratic APN function, x3 +Tr(x9), showing that its Walsh transform is 3-valued for odd n, and is 5-valued for even n. Therefore, the distribution of the values of the Walsh transform of x3 + Tr(x9) is the same as that of the APN Gold functions. Moreover, for odd n the function is AB, which gives an alternative proof of the APN property of the function.

We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.