$C$ is linearly equivalent to a Hermitian self-orthogonal code if and only if there are non-zero $\lambda_i \in {\mathbb F}_{q}$ such that $$ \sum_{i=1}^n \lambda_{i} u_{i}v_{i}^q=0, $$ for all $u,v \in C$.

For any linear code $C$ over ${\mathbb F}_{q^2}$ of length $n$, Rains defined the puncture code $P(C)$ to be $$ P(C)=\{\lambda=(\lambda_{1},\ldots,\lambda_n) \in {\mathbb F}_q^n \ | \ \sum_{{i}=1}^n \lambda_{i} u_{i}v_{i}^q=0, \ \mathrm{for} \ \mathrm{all} \ u,v \in C \}. $$

$C$ has a truncation of length $r \leqslant n$ which is linearly equivalent to a Hermitian self-orthogonal code if and only if there is an element of $P(C)$ of weight $r$.

Rains was motivated to look for Hermitian self-orthogonal codes, since there is a simple way to construct a $ [\![ n,n-2k]\!] _q$ quantum code, given a Hermitian self-orthogonal code.

In this talk, I will detail an effective way to calculate the puncture code. I will outline how to prove various results about when a linear code has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code and how to extend it to one that does in the case that it has no such truncation. In the case that the code is a Reed-Solomon code, it turns out that the existence of such a truncation of length $r$ is equivalent to the existence of a polynomial $g(X)\in {\mathbb F}_{q^2}[X]$ of degree at most $(q-k)q-1$ with the property that $g(X)+g(X)^q$ has $q^2-r$ distinct zeros in ${\mathbb F}_{q^2}$.

In this talk we give a brief overview of the cross-correlation between m-sequences and some applications. Furthermore, we show how some problems on m-sequences are related to problems in other areas of mathematics such as difference sets, strongly regular graphs, almost perfect nonlinear functions, solutions of equations over finite fields.

In this work, we consider more general digital signature schemes which we call modification-tolerant signature schemes, which go beyond the ability of detecting modifications, and have the ability of locating modifications or locating and correcting modifications. They can be used in applications where either the data can be modified (collaborative work), or the data must be modified (redactable and content extraction signatures) or we need to know which parts of the data have been modified (data forensics).

We discuss two types of modification-tolerant signature schemes: a general one that allows the location of modified blocks of the data, and a scheme with correction capability, that allows the correction of the modified blocks, recovering the original message. We give three instantiations of the scheme for the purpose of location, correction, and redaction. The schemes are proposed using techniques from combinatorial group testing.

This talk is based on joint work with Lucia Moura and Carlisle Adams.

The talk is based on joint works with Pascale Charpin and Lukas Kölsch, Björn Kriepke.

Joint work with Dunia Marchiori, Ricardo Custodio and Lucia Moura.

Three recent generalizations of AONTs are motivated by applications due to Pham et al. and Oliveira et al. We term these generalizations rectangular, range, and restricted AONTs. Briefly, in a rectangular AONT, the number of outputs is greater than the number of inputs. A range AONT satisfies the $t$-AONT property for a range of consecutive values of $t$. Finally, in a restricted AONT, the unknown outputs are assumed to occur within a specified set of "secure" output blocks. We study existence and non-existence and provide examples and constructions for these generalizations. We also demonstrate interesting connections with combinatorial structures such as orthogonal arrays, split orthogonal arrays, MDS codes and difference matrices.

This talk is based on joint work with Navid Nasr Esfahani.

However, such a linear code is only interesting if the $p$-rank of the point-block incidence matrix is small enough. Hamada (1973) determined the $p$-rank for the incidence matrices of so-called classical or geometric designs. It is a long-standing conjecture that the incidence matrices from this class of designs are of minimal $p$-rank.

Dela Cruz, Wassermann (2021) showed that linear codes from subspace designs ($q$-analogues of combinatorial designs) have the same decoding properties as the linear codes from their corresponding geometric designs, but for many parameters the majority logic decoder needs exponentially less parity check equations.

In this talk, we will show that Hamada's formula can also be applied to $q$-analogues of group divisible designs and lifted MRD codes ($q$-analogues of transversal designs), which may make these classes of codes interesting.

Today, majority logic decodable codes are still interesting for devices with low computational power and because of the relation to linear locally repairable codes and private information retrieval (PIR) codes.

It is a bit surprising that the existence of an Abelian-Cayley Moore graph is equivalent to a perfect linear Lee code of radius $r$ in $\mathbb{Z}^n$, that is a lattice tiling of $\mathbb{Z}^n$ by the translations of an $\ell_1$-metric sphere of radius $r$. More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee code $C$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the speaker proved that if $C$ is linear, then Golomb-Welch conjecture is true for $r=2$ and $n\geq 3$.

In this talk, we consider the classification of linear Lee codes of the second best possibility, that is the density of the lattice packing of $\mathbb{Z}^n$ by Lee spheres $S(n,r)$ equals $\frac{|S(n,r)|}{|S(n,r)|+1}$. By checking the corresponding abelian Cayley graphs, an almost perfect linear Lee code is equivalent to the case with $G=\Psi \cup \{f\}$ where $f$ is the unique element of order $2$ in $G$.