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Anuradha S. Garge

Publications and source records attributed to Anuradha S. Garge.

6 recordsLinked to original sources
arXiv2026

Suslin-Rao Codes : the bridge between pure and applied

In the year $1977,$ while developing the theory of modules arising in the context of Serre's problem on projective modules in the paper ``On Stably Free Modules" (see \cite{AS}), A. A. Suslin constructed matrices inductively using unimodular rows. Our aim is to connect these matrices to the current and applicable field of coding theory. To this end, we construct Suslin-Rao codes, by using looking at certain unimodular all one vectors over the finite field $\mathbb{F}_2$. Comparative analysis with standard families such as Hamming, Simplex and BCH codes shows that the Suslin-Rao codes achieve better performance than Hamming and Simplex codes in terms of code rate and error correction capability, though BCH codes may outperform them for large code lengths. Bit error rate (BER) versus signal-to-noise ratio (SNR) simulations further support the practical applicability of these codes in noisy channels. The recursive nature of the construction allows efficient encoder design, making the proposed codes attractive for real-time communication systems and hardware implementation.

cs.IT↗
arXiv2026

Integral quadratic forms over a ring of $p$-adic integers

Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \mathbb{Z}$, $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z})$. He gave bounds on $f(n)$ too. Koo and Lee further improved the bounds in 2025. This paper is organized as follows. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. In the first section we give necessary and sufficient condition for a quadratic form $\sum_{i=1}^{m}a_iX_i^2$ to be universal over $M_2(\mathbb{Z}_p)$ for $p=2$ and for an odd prime $p$. Consequently we express matrices of over $\mathbb{Z}_p$ as sum of squares. For a positive integer $n \geq 3$, we define $f(n)$ to be the smallest positive integer $m$ such that for every $p$-adic integers $a_1,a_2, \ldots a_m$ with at least three of them units, the diagonal quadratic form $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z}_p)$. In the next section we find bounds on $f(n)$ for $n \geq 3$.

math.NT↗
arXiv2026

On the Algebraic Structure Underlying the Support Enumerators of Linear Codes

In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block codes. More precisely, we have established formula for counting codewords in the linear code C whose i-th coordinate is nonzero. Moreover, we derived a MacWilliam's type identity, relating the normalized support enumerators of a linear code and its dual, explaining how coordinate information transforms under duality. Using this identity we deduce a condition for self duality based on the equality of support distributions. These results provide a more detailed understanding of code structure and complement classical weight based duality theory.

cs.IT↗
arXiv2020

Trace and discriminant criteria for a matrix to be a sum of sixth and eighth powers of matrices

In this paper, we shall be considering the Waring's problem for matrices. One version of the problem involves writing an $n \times n$ matrix over a commutative ring $R$ with unity as a sum of $k$-th powers of matrices over $R.$ This study is motivated by the interesting results of Carlitz, Newman, Vaserstein, Griffin, Krusemeyer, Richman etc. obtained earlier in this direction. The results are for the case $n \geq k \geq 2$ in terms of the trace of the matrix. For $n < k,$ it was shown by Katre, Garge that it is enough to work with the special case $n = 2$ and $k \geq 3.$ The cases $3 \leq k \leq 5$ and $k = 7$ were settled in earlier results. There was no case of a composite, non-prime-power $k$ occuring above. In this paper, we will find the trace criteria for a matrix to be a sum of sixth (a composite non-prime power) and eighth powers of matrices over a commutative ring $R$ with unity. An elegant discriminant criterion was obtained by Katre and Khule earlier in the special case of an order in an algebraic number field $\mathcal{O}.$ We will derive here similar discriminant criteria for every matrix over $\mathcal{O}$ to be a sum of sixth and eighth powers of matrices over $\mathcal{O}.$

math.NT↗
arXiv2013

Global actions, K-theory and unimodular rows

Global actions were introduced by Bak in order to have a homotopy theory in a purely algebraic setting. In this paper we apply his techniques in a particular case: the (single domain) unimodular row global action. More precisely, we compute the the path connected component and fundamental group for the unimodular row global action. An explicit computation of the fundamental group of the (connected component of) unimodular row global action is closely related to stability questions in K-theory. This will be shown by constructing an exact sequence with the fundamental group functor as the middle term and having surjective stability for the functor K_2 on the left and injective stability for the functor K_1 on the right.

math.KT↗