In this paper, we bound the rate of linear codes in Fnqwith the property that anyk≤qcodewords are all simultaneously distinct in at leastdkcoordinates. For the case of particular interestq=k= 3 we recover, with a simpler proof, state of the art results in the cased3= 1 and new bounds ford3> 1. We finally discuss some related open problems on the list-decoding zero-error capacity of discrete memoryless channels.

The incorporation of LiDAR technology into some high-end smartphones has unlocked numerous possibilities across various applications, including photography, image restoration, augmented reality, and more. In this paper, we introduce a novel direction that harnesses LiDAR depth maps to enhance the compression of the corresponding RGB camera images. To the best of our knowledge, this represents the initial exploration in this particular research direction. Specifically, we propose a Transformer-based learned image compression system capable of achieving variable-rate compression using a single model while utilizing the LiDAR depth map as supplementary information for both the encoding and decoding processes. Experimental results demonstrate that integrating LiDAR yields an average PSNR gain of 0.83 dB and an average bitrate reduction of 16% as compared to its absence.

This paper presents new upper bounds on the rate of linear k-hash codes in$\mathbb{F}_{q}^{n}, q\geq k$, that is, codes with the property that any$k$distinct codewords are all simultaneously distinct in at least one coordinate.

In this paper, we study upper bounds on the minimum length of frameproof codes introduced by Boneh and Shaw [3] to protect copyrighted materials. A q-ary (k,n)-frameproof code of length t is a t×n matrix having entries in {0,1,…,q−1} and with the property that for any column c and any other k columns, there exists a row where the symbols of the k columns are all different from the corresponding symbol (in the same row) of the column c. In this paper, we show the existence of q-ary (k,n)-frameproof codes of length $t = O\left( {\frac{{{k^2}}}{q}\log n} \right)$ for q ≤ k, using the Lovász Local Lemma, and of length $t = O\left( {\frac{{{k^2}}}{{\log \left( {q/k} \right)}}\log \left( {n/k} \right)} \right)$ for q > k using the expurgation method. Remarkably, for the practical case of q ≤ k our findings give codes whose length almost matches the lower bound $\Omega \left( {\frac{{{k^2}}}{{q\log k\log n}}} \right)$ on the length of any q-ary (k,n)-frameproof code and, more importantly, allow us to derive an algorithm of complexity O(tn2) for the construction of such codes.

In this paper, we study bounds on the minimum length of ( k, n, d)-superimposed codes introduced by Agarwal et al. [1], in the context of Non-Adaptive Group Testing algorithms with runlength constraints. A ( k, n, d)-superimposed code of length t is a t × n binary matrix such that any two 1’s in each column are separated by a run of at least d 0’s, and such that for any column c and any other k−1 columns, there exists a row where c has 1 and all the remaining k−1 columns have 0. Agarwal et al. proved the existence of such codes with t = Θ( dk log( n/k) + k2log( n/k)). Here we investigate more in detail the coefficients in front of these two main terms as well as the role of lower order terms. We show that improvements can be obtained over the construction in [1] by using different constructions and by an appropriate exploitation of the Lovász Local Lemma in this context. Our findings also suggest O( nk) randomized Las Vegas algorithms for the construction of such codes. We also extend our results to Two-Stage Group Testing algorithms with runlength constraints.

For fixed integers$n$and$b\geq k$, let$A(b,k,n)$the largest size of a subset of$\{1,2,\ldots,b\}^{n}$such that, for any$k$distinct elements in the set, there is a coordinate where they all differ. Bounding$A(b,k,n)$is a problem of relevant interest in information theory and computer science, relating to the zero-error capacity with list decoding and to the study of$(b, k)$-hash families of functions. It is known that, for fixed$b$and$k$,$A(b,k,n)$grows exponentially in$n$. In this paper, we determine new exponential upper bounds for different values of$b$and$k$. A first bound on$A(b,k,n)$for general$b$and$k$was derived by Fredman and Komlós in the ’80s and improved for certain$b\neq k$by Körner and Marton and by Arikan. Only very recently better bounds were derived for general$b$and$k$by Guruswami and Riazanov, while stronger results for small values of$b=k$were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan, and by Costa and Dalai. In this paper, we strengthen the bounds for some specific values of$b$and$k$. Our contribution is a new computational method for obtaining upper bounds on the values of a quadratic form defined over discrete probability distributions in arbitrary dimensions, which emerged as a central ingredient in recent works. The proposed method reduces an infinite-dimensional problem to a finite one, which we manage to further simplify by means of a series of optimality conditions.

For fixed integers$b\geq k$, the problem of perfect$(b,\ k)$-hashing asks for the asymptotic growth of largest subsets of$\{1, 2, \ldots, b\}^{n}$such that for any$k$distinct elements in the set, there is a coordinate where they all differ. An important asymptotic upper bound for general, was derived by Fredman and Komlós in the ‘80s and improved for certain by Körner and Marton and by Arikan. Only very recently better bounds were derived for the general case by Guruswami and Riazanov, while stronger results for small values of were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Costa and Dalai. In this paper, we both show how some of the latter results extend to and further strengthen the bounds for some specific small values of and. The method we use, which depends on the reduction of an optimization problem to a finite number of cases, shows that further results might be obtained by refined arguments at the expense of higher complexity.