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B. A. Sethuraman

Publications and source records attributed to B. A. Sethuraman.

8 recordsLinked to original sources
arXiv2014

Counterexample to the $l$-modular Belfiore-Solé Conjecture

We show that the secrecy function conjecture that states that the maximum of the secrecy function of an $l$-modular lattice occurs at $1/\sqrt{l}$ is false, by proving that the 4-modular lattice $C^(4) = \mathbb{Z} \oplus \sqrt{2}\mathbb{Z} \oplus 2\mathbb{Z}$ fails to satisfy this conjecture. We also indicate how the secrecy function must be modified in the $l$-modular case to have a more reasonable chance for it to have a maximum at $1/\sqrt{l}$, and show that the conjecture, modified with this new secrecy function, is true for various odd 2-modular lattices.

cs.IT↗
arXiv2014

Bounds of fast decodability of space time block codes, skew-Hermitian matrices, and Azumaya algebras

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 $l$.

cs.IT↗
arXiv2013

Hilbert series of certain jet schemes of determinantal varieties

We consider the affine variety ${\mathcal{Z}_{2,2}^{m,n}}$ (or just "$Y$") of first order jets over ${\mathcal{Z}_{2}^{m,n}}$ (or just "$X$"), where $X$ is the classical determinantal variety given by the vanishing of all $2\times 2$ minors of a generic $m\times n$ matrix. When $2 < m \le n$, this jet scheme $Y$ has two irreducible components: a trivial component, isomorphic to an affine space, and a nontrivial component that is the closure of the jets supported over the smooth locus of $X$. This second component is referred to as the principal component of $Y$; it is, in fact, a cone and can also be regarded as a projective subvariety of $\mathbf{P}^{2mn-1}$. We prove that the degree of the principal component of $Y$ is the square of the degree of $X$ and more generally, the Hilbert series of the principal component of $Y$ is the square of the Hilbert series of $X$. As an application, we compute the $a$-invariant of the principal component of $Y$ and show that the principal component of $Y$ is Gorenstein if and only if $m=n$.

math.CO↗
arXiv2012

Quotients of Orders in Cyclic Algebras and Space-Time Codes

Let $F$ be a number field with ring of integers $\Oc_F$ and $\Dc$ a division $F$-algebra with a maximal cyclic subfield $K$. We study rings occurring as quotients of a natural $\Oc_F$-order $Λ$ in $\Dc$ by two-sided ideals. We reduce the problem to studying the ideal structure of $Λ/\qf^sΛ$, where $\qf$ is a prime ideal in $\Oc_F$, $s\geq 1$. We study the case where $\qf$ remains unramified in $K$, both when $s=1$ and $s>1$. This work is motivated by its applications to space-time coded modulation.

cs.IT↗
arXiv2009

Jet Schemes of the Commuting Matrix Pairs Scheme

We show that for all $k\ge 1$, there exists an integer $N(k)$ such that for all $n\ge N(k)$ the $k$-th order jet scheme over the commuting $n\times n$ matrix pairs scheme is reducible. At the other end of the spectrum, it is known that for all $k\ge 1$, the $k$-th order jet scheme over the commuting $2\times 2$ matrices is irreducible: we show that the same holds for $n=3$.

math.AG↗
arXiv2005

Perfect Space-Time Codes with Minimum and Non-Minimum Delay for Any Number of Antennas

Perfect space-time codes were first introduced by Oggier et. al. to be the space-time codes that have full rate, full diversity-gain, non-vanishing determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping of the constellation. These defining conditions jointly correspond to optimality with respect to the Zheng-Tse D-MG tradeoff, independent of channel statistics, as well as to near optimality in maximizing mutual information. All the above traits endow the code with error performance that is currently unmatched. Yet perfect space-time codes have been constructed only for 2,3,4 and 6 transmit antennas. We construct minimum and non-minimum delay perfect codes for all channel dimensions.

cs.IT↗
arXiv2002

Determinantal Varieties Over Truncated Polynomial Rings

We study higher order determinantal varieties obtained by considering generic $m\times n$ ($m \le n$) matrices over rings of the form $F[t]/(t^k)$, and for some fixed $r$, setting the coefficients of powers of $t$ of all $r \times r$ minors to zero. These varieties can be interpreted as generalized tangent bundles over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when $r = m$, the varieties are irreducible, but when $r < m$, these varieties have at least $\lfloor {k/2}\rfloor + 1$ components. In fact, when $r=2$ (for any $k$), or when $k=2$ (for any $r$), there are exactly $\lfloor {k/2}\rfloor + 1$ components. We give formulas for the dimensions of these components in terms of $k$, $m$, and $n$. In the case of square matrices with $r=m$, we show that the ideals of our varieties are prime and that the coordinate rings are complete intersection rings, and we compute the degree of our varieties via the combinatorics of a suitable simplicial complex.

math.AG↗