We introduce two constructions of additive codes over finite fields. Both constructions start with a linear code over a field withqelements and give additive codes over the field withqhelements whose minimum distance is demonstrably good.

Purity testing protocols (PTPs), also known as Bell state testing protocols, i.e., protocols that decide with high probability whether or not a distributed bipartite quantum state is a tensor product of Bell states, have been proven to be a useful tool in many quantum communication applications. In this paper, we provide geometrical constructions for such protocols that originate directly from classical linear error correcting codes (LECCs), in a way that the properties of the resulting PTP are completely determined from those of the LECC used in the construction. We investigate the implications of our results in various quantum communication tasks, including error detection, entanglement purification for general quantum error models and quantum message authentication.

Abstract We consider the cyclic presentation of $$\textrm{PG}(3, q )$$ PG ( 3 , q ) whose points are in the finite field $$\mathbb {F}_{q^4}$$ F q 4 and describe the known ovoids therein. We revisit the set $$\mathcal {O}$$ O , consisting of $$(q^2+1)$$ ( q 2 + 1 ) th roots of unity in $$\mathbb {F}_{q^4}$$ F q 4 , and prove that it forms an elliptic quadric within the cyclic presentation of $$\textrm{PG}(3, q )$$ PG ( 3 , q ) . Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki–Tits ovoids in the cyclic presentation of $$\textrm{PG}(3, q )$$ PG ( 3 , q ) , characterizing them as the zeroes of a polynomial over $$\mathbb {F}_{q^4}$$ F q 4 .

We prove that the natural isomorphism$\mathbb {F}_{2^{h}}\cong \mathbb {F} _{2}^{h}$induces a bijection between stabilizer codes on n quqits with local dimension$q=2^{h}$and binary stabilizer codes on hn qubits. This allows us to describe these codes geometrically: a stabilizer code over a field of even order corresponds to a so-called quantum set of symplectic polar spaces. Moreover, equivalent stabilizer codes have a similar geometry, which can be used to prove the uniqueness of a$[\![{4,0,3}]\!]_{4}$stabilizer code and the nonexistence of both a$[\![{7,1,4}]\!]_{4}$and an$[\![{8,0,5}]\!]_{4}$stabilizer code.

We prove that there is a Hermitian self-orthogonal$k$-dimensional truncated generalised Reed-Solomon code of length$n \leqslant q^{2}$over${\mathbb F}_{q^{2}}$if and only if there is a polynomial$g \in {\mathbb F}_{q^{2}}$of degree at most$(q-k)q-1$such that$g+g^{q}$has$q^{2}-n$distinct zeros. This allows us to determine the smallest$n$for which there is a Hermitian self-orthogonal$k$-dimensional truncated generalised Reed-Solomon code of length$n$over${\mathbb F}_{q^{2}}$, verifying a conjecture of Grassl and Rötteler. We also provide examples of Hermitian self-orthogonal$k$-dimensional generalised Reed-Solomon codes of length$q^{2}+1$over${\mathbb F}_{q^{2}}$, for$k=q-1$and$q$an odd power of two.

A normal rational curve of the (k - 1)-dimensional projective space over Fq is an arc of size q+1, since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k -6 of a normal rational curve cannot be extended to an arc of size q +2. In fact, we prove something slightly stronger. Suppose that q is odd and E is a (2k - 3)-subset of an arc G of size 3k - 6. If G projects to a subset of a conic from every (k - 3)-subset of E, then G cannot be extended to an arc of size q + 2. Stated in terms of errorcorrecting codes we prove that a k-dimensional linear maximum distance separable code of length 3k - 6 over a field Fq of odd characteristic, which can be extended to a Reed-Solomon code of length q +1, cannot be extended to a linear maximum distance separable code of length q + 2.

It is shown that the parameters of a linear code over ${\mathbb F}_q$ of length $n$, dimension $k$, minimum weight $d$, and maximum weight $m$ satisfy a certain congruence relation. In the case that $q=p$ is a prime, this leads to the bound $m \leq (n-d)p-e(p-1)$, where $e \in \{0,1,\ldots,k-2 \}$ is maximal with the property that ${n-d \choose e} \not\equiv 0 \pmod{p^{k-1-e}}.$ Thus, if $C$ contains a codeword of weight $n$, then $n \geq d/(p-1)+d+e$. The results obtained for linear codes are translated into corresponding results for $(n,t)$-arcs and $t$-fold blocking sets of AG$(k-1,q)$. The bounds obtained in these spaces are better than the known bounds for these geometrical objects for many parameters.

Every biuniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given biuniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all biuniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of biuniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields.