On additive MDS codes with linear projections

Technical Summary of "On additive MDS codes with linear projections"

Problem Statement
The paper investigates the structural properties of additive Maximum Distance Separable (MDS) codes over finite fields. Specifically, it addresses the question of whether long additive MDS codes are necessarily equivalent to linear codes. While the MDS conjecture posits that for linear codes, the length $n$ is bounded by $q+1$ (with specific exceptions), the behavior of additive codes (which are linear over a subfield but not necessarily over the full field) is less understood. The authors aim to provide evidence supporting the hypothesis that sufficiently long additive MDS codes over $\mathbb{F}_{q^h}$, which possess certain "linear-like" projections, must themselves be equivalent to linear codes (or at least linear over a larger subfield).

Methodology and Framework
The authors utilize a geometric approach established in prior work [BGL22], linking additive codes to geometric objects known as pseudo-arcs.

1. Geometric Correspondence: An $\mathbb{F}_q$-linear code $C$ of length $n$ and dimension $k$ over $\mathbb{F}_{q^h}$ corresponds to a projective $h$-system in $PG(k-1, q)$. The code is MDS if and only if this system is a pseudo-arc (a set of subspaces where any $k$ span the space).
2. Linear Equivalence Criterion: A code is equivalent to a linear code if and only if its associated projective $h$-system is contained within a Desarguesian spread.
3. Projection Analysis: The core method involves analyzing the projections of the code (and the corresponding pseudo-arc) from specific coordinate positions. The authors leverage known results regarding pseudo-arcs and translation generalised quadrangles (e.g., results by Penttila, Van de Voorde, and others) which state that if projections of a pseudo-arc lie in a Desarguesian spread, the arc itself often does as well.
4. Algebraic Characterization: For the proofs, the authors express the MDS code in a "standard form" defined by an $\mathbb{F}_q$-linear map $f$. They analyze the conditions under which the projections being linear forces the defining polynomials of the code to be monomials or semi-linear, thereby implying the code's linearity.

Key Contributions and Results
The paper establishes two main theorems providing sufficient conditions for an additive MDS code to be equivalent to a linear code:

• Theorem 1.1 (Case $k=3$):
  Let $C$ be an $\mathbb{F}_q$-linear $(n, q^{3h}, n-2)_{q^h}$ MDS code. If $n$ is sufficiently large (specifically $n \ge 2q+3$ for $h=2$, and $n \ge q^2+3+\delta_{2,q}$ for $h=3$) and $C$ has three coordinates such that the projections from these coordinates are equivalent to linear codes, then $C$ itself is equivalent to a linear code.

  • Proof Strategy: This relies on Lemma 5.10, which reduces the problem to showing that pairs of invertible linearised polynomials $(f, g)$ satisfying a specific property (Prop$_m$) must be monomials. The proof uses Dickson matrices and rank arguments to show that for large $m$, the polynomials must have zero coefficients, forcing them to be monomials.

• Theorem 1.2 (Case $k>3$):
  Let $C$ be an $\mathbb{F}_q$-linear $(n, q^{hk}, n-k+1)_{q^h}$ MDS code with $k > 3$. If there exist two disjoint subsets of coordinates $A$ and $B$ such that $|A| + |B| \le k-2$, and the projections of $C$ from $A$ and $B$ are equivalent to linear codes, and $n > q+k$, then $C$ is equivalent to a code linear over a larger field $\mathbb{F}_{q^s}$ (where $s$ is a divisor of $h$).

  • Refinement: If $n$ is even larger ($n > q^e + k$, where $e$ is the largest proper divisor of $h$), then $C$ is equivalent to a linear code over $\mathbb{F}_{q^h}$.
  • Proof Strategy: The proof constructs a standard form code and uses the linearity of the projections to derive constraints on the defining polynomials. It demonstrates that the existence of two disjoint sets of linear projections forces the defining map to be semi-linear over a subfield, eventually implying full linearity under the length constraints.

Significance and Claims
The authors frame these results as providing "some evidence" for the broader conjecture that long additive MDS codes are equivalent to linear codes.

• Bridging Geometry and Coding Theory: The paper successfully translates geometric results about pseudo-arcs and Desarguesian spreads (specifically Result 4.4 from [BGL22]) into concrete coding-theoretic theorems regarding additive MDS codes.
• Extension of Known Results: The work extends previous findings (such as Result 4.6) which were limited to specific cases (e.g., $h=2$). Theorem 1.1 generalizes these to $h=3$ and provides a unified framework for $k=3$.
• Tightness of Bounds: The paper acknowledges that the bounds on $n$ in the theorems are likely not tight. It provides a counter-example construction (Lemma 5.8) showing that if the length $n$ is too small (specifically near the bound $n_q(k)$), one can construct non-linear additive MDS codes that still possess linear projections. This highlights the necessity of the "long code" assumption in their theorems.

The paper concludes by noting that while the results support the conjecture, the existence of pseudo-arcs not contained in Desarguesian spreads remains an open geometric question, and the algebraic proofs rely on specific bounds for the length of the codes.