Title: Algebraic Geometry codes in the sum-rank metric
Domain: Math

Question: How can one construct the first geometric codes in the sum-rank metric, known as linearized Algebraic Geometry codes, using Ore polynomial rings with coefficients from algebraic curve function fields, and how can this construction surpass existing performance bounds? Specifically, how are quotient rings of Ore polynomials defined, how does the sum-rank metric code structure work with evaluation maps, and what are the bounds on the dimension and minimum distance of these codes compared to classical bounds like Goppa’s and Gilbert–Varshamov?

Background: | 1. In the Hamming metric, Reed–Solomon codes achieve optimal trade-offs between length, dimension, and distance, but their construction requires large finite fields when long codes are needed. | 2. In the rank and sum-rank metrics, Gabidulin codes and linearized Reed–Solomon codes play analogous roles, but they inherit the same limitation: long codes cannot be constructed without dramatically enlarging the field size. | 3. Algebraic Geometry codes in the Hamming setting provided a breakthrough by evaluating functions on algebraic curves with many rational points, allowing longer codes over smaller fields and even producing asymptotic improvements over random coding bounds. | 4. In contrast, for the rank and sum-rank metrics, no geometric constructions had been available, so there was no analogue of the rich curve-based methods known in the Hamming case. This left open the problem of finding codes in these metrics that combine structural elegance with strong asymptotic parameters. |

Definitions: | 1. Ore Polynomial Rings: Noncommutative rings consisting of polynomials \(L[T;\Phi]\), where multiplication is twisted by an automorphism \(\Phi\). Quotients of the form \(L[T;\Phi]/(T^r - x)\) provide a natural algebraic environment in which to encode curve-based information. | 2. Noncommutative Riemann–Roch Spaces: Vector spaces of Ore polynomial elements constrained by valuations relative to divisors on algebraic curves. They generalize the classical Riemann–Roch spaces and allow one to transport divisor theory into the noncommutative setting. | 3. Sum-Rank Metric: A distance measure defined on tuples of linear maps or matrices. The weight of a tuple is the sum of the individual ranks, and the code distance is induced from this weight. | 4. Linearized Algebraic Geometry Codes: Codes constructed by evaluating elements from noncommutative Riemann–Roch spaces at selected places of a curve, producing tuples of endomorphisms. These codes generalize Reed–Solomon codes (Hamming metric) and Gabidulin codes (rank metric) into the sum-rank setting. |

Methodology: | 1. Algebraic Tools from Ore Rings: Properties of Ore polynomial quotients ensure that evaluation maps are well-defined and injective, avoiding zero divisors and guaranteeing consistent code construction. | 2. Dimension Bounds via Riemann–Roch: A generalized Riemann–Roch theorem applied in the noncommutative framework provides lower bounds on the dimension of code spaces. | 3. Distance Analysis through Evaluation Maps: Studying kernels of evaluation morphisms and reduced norm arguments establishes lower bounds on the minimum distance of codewords. | 4. Asymptotic Constructions from Curves: Using towers of algebraic curves with many rational points allows the formation of infinite sequences of codes whose parameters eventually beat the Gilbert–Varshamov bound in the sum-rank metric. |

Answer: The theory of **linearized Algebraic Geometry (AG) codes in the sum-rank metric** introduces a new class of codes that extend the rich interplay between algebraic geometry and coding theory into a noncommutative framework. These codes arise from combining **Ore polynomial rings** with the function fields of algebraic curves, thereby generalizing both classical AG codes (in the Hamming metric) and linearized Reed–Solomon or Gabidulin codes (in rank and sum-rank metrics). Their construction not only provides explicit new families of codes but also yields parameter bounds that surpass existing performance thresholds, including asymptotic improvements over the Gilbert–Varshamov (GV) bound. | | --- | | ## 1. Ore Polynomial Rings and Noncommutative Riemann–Roch Theory | | The foundation of this construction lies in **Ore polynomial rings**. Given a cyclic Galois extension $L/K$ of function fields of curves with automorphism $\Phi$, one considers the Ore polynomial ring $L[T;\Phi]$, where multiplication is twisted by the relation | | $$ | T \cdot a = \Phi(a) T, \quad a \in L. | $$ | | Taking the quotient by relations of the form $T^r - x$, with $x \in K^\times$, yields algebras $D_{L,x} = L[T;\Phi]/(T^r - x)$. These are central simple algebras that behave as noncommutative analogues of function fields. | | Within this setting, one extends **divisor theory** and the **Riemann–Roch framework**. For each divisor on a covering curve $Y$ lying above a base curve $X$, one defines a Riemann–Roch space $\Lambda_{L,x}(E)$ consisting of Ore polynomial elements subject to valuation constraints at places of $Y$. These spaces are finite-dimensional over the base field and admit explicit lower bounds on their dimensions, closely paralleling the classical Riemann–Roch theorem. Thus, the classical geometric machinery of divisors and function spaces is successfully transplanted into the noncommutative Ore polynomial environment. | | --- | | ## 2. Sum-Rank Metric and Evaluation Mechanism | | The **sum-rank metric** interpolates between Hamming and rank metrics. Given vector space tuples $W = (W_1,\dots,W_s)$ and $V = (V_1,\dots,V_s)$, the ambient space | | $$ | \text{Hom}_k(W,V) = \prod_{i=1}^s \text{Hom}_k(W_i, V_i) | $$ | | carries the sum-rank weight, defined as the sum of the ranks of each linear map. A code in this metric is a linear subspace of this ambient space. | | The construction of linearized AG codes proceeds by defining **evaluation maps**. Fixing rational places on the base curve, one evaluates Ore polynomial functions in the Riemann–Roch space at selected points, producing tuples of linear maps in the ambient space. These evaluations, reduced modulo appropriate local parameters, yield matrices that form the codewords. This mechanism is the natural generalization of evaluating rational functions at rational points to obtain classical AG codes, but now within a noncommutative algebraic framework. | | --- | | ## 3. Explicit Construction and Hypotheses for Well-Defined Codes | | A **linearized Algebraic Geometry code** is defined as the image of the multi-evaluation map | | $$ | \alpha : \Lambda_{L,x}(E) \to \prod_{i=1}^s \text{End}_k(V_i), | $$ | | where $E$ is a chosen divisor on the covering curve, and the $V_i$ are local quotient spaces at evaluation places. Several hypotheses guarantee that this construction is well-defined: | | * The algebra $D_{L,x}$ must contain no zero divisors, ensuring injectivity of evaluation. | * Compatibility conditions at evaluation places must hold so that Ore elements can be consistently reduced. | | Under these assumptions, the codes are rigorously defined, and their parameters can be studied with precision. | | --- | | ## 4. Parameter Bounds: Dimension and Minimum Distance | | The central result concerns **parameter bounds**. For a code $C$ constructed from divisor $E$, one obtains: | | * A **dimension bound**: | | $$ | \dim C \;\geq\; r \cdot \deg(E) - r \cdot (g_Y - 1) - \text{local correction terms}, | $$ | | where $g_Y$ is the genus of the covering curve $Y$, and the correction terms depend on ramification and local invariants. | | * A **minimum distance bound**: | | $$ | d \;\geq\; sr - \deg(E), | $$ | | where $s$ is the number of evaluation places and $r$ the extension degree. | | These bounds generalize Goppa’s bounds for classical AG codes in the Hamming metric. Importantly, the codes simultaneously exhibit large dimension and large minimum distance, a hallmark of good code families. | | --- | | ## 5. Comparison with Classical Bounds | | The parameters of linearized AG codes satisfy an analogue of the Singleton inequality in the sum-rank metric. More significantly, in appropriate field settings—such as finite fields of square cardinality—these codes yield families that **asymptotically surpass the Gilbert–Varshamov bound** in the sum-rank metric. This result mirrors the landmark Tsfasman–Vlăduţ–Zink breakthrough for AG codes in the Hamming metric, establishing that geometry-based constructions can again outperform random coding bounds in a broader metric context. | | --- | | ## 6. Relation to Reed–Solomon and Gabidulin Codes | | Linearized AG codes recover known families as special cases: | | * For the projective line as the base curve, the construction reduces to **linearized Reed–Solomon codes**, which are maximum sum-rank distance (MSRD) codes attaining the Singleton bound. | * By varying evaluation, one recovers **Gabidulin codes**, the classical rank-metric analogue of Reed–Solomon codes. | | Compared to these, linearized AG codes provide **longer lengths** over the same finite fields (thanks to curves with many rational points) and require **smaller field sizes** for given dimensions and distances. This overcomes the key limitation of Reed–Solomon-type codes, which demand large field extensions for long code lengths. | | --- | | ## 7. Asymptotic Goodness and Beyond | | By considering towers of curves with many rational points, one constructs infinite families of linearized AG codes with asymptotically positive rate and relative distance, hence proving their **asymptotic goodness**. Even stronger, when the field size is sufficiently large, these codes strictly improve on the random coding GV bound in the sum-rank setting. This demonstrates that geometry-driven methods remain a source of asymptotically optimal constructions in modern coding theory. | | --- | | ## 8. Broader Perspective | | The emergence of linearized Algebraic Geometry codes marks a conceptual advance: | | * They extend AG code theory to noncommutative algebras (Ore rings). | * They unify and generalize classical Reed–Solomon and Gabidulin families. | * They provide explicit and implementable constructions with strong parameter guarantees. | * They surpass probabilistic bounds asymptotically, reaffirming the power of algebraic geometry in coding. | | This development thus positions linearized AG codes as a central object for future research, bridging noncommutative algebra, algebraic geometry, and information theory. | | --- | | |

Checklist: 1. Defines the algebraic setup with integers \(r,d,m\) (where \(r=md\)), the Galois extension \(L_0/K\), the construction of \(L\), the automorphism \(\Phi\), the Ore polynomial ring \(L[T;\Phi]\), the quotient algebra \(D_{L,x}\), and the reduced norm map. | | 2. States the Laurent series framework: \(O_K=k[[t]], K=k((t))\), the extension of \(v_t\) to \(L_0\), the valuation maps \(w_{j,x}\), the subring \(\Lambda_{L,x}\), and the conditions for \(\gamma_u\) to induce homomorphisms. | | 3. Describes the algebraic curve background: the function field \(K=k(X)\), divisor group \(\mathrm{Div}(X)\), Riemann–Roch spaces, the Galois cover \(\pi:Y\to X\), valuation extensions to \(Y\), and the definition of \(\rho_p\). | | 4. Defines the noncommutative Riemann–Roch space \(\Lambda_{L,x}(E)\), its decomposition into classical spaces \(L_Y(E_i)\), and the derivation of the dimension lower bound. | | 5. States the prerequisites for code construction: rational places \(p_1,\dots,p_s\), the zero-divisor-free condition for \(D_{L,x}\), the norm hypotheses at places, and the setup of the ambient space \(H=\prod_i \mathrm{End}_k(V_i)\). | | 6. Defines the evaluation map \(\alpha:\Lambda_{L,x}(E)\to H\) via the homomorphisms \(\gamma_u\), \(\varepsilon\), and reductions, and introduces the code \(C=\mathrm{Im}(\alpha)\). | | 7. Proves injectivity of \(\alpha\) under the zero-divisor-free assumption, derives the minimum distance bound \(d\ge sr-\deg_Y(E)\), the dimension bound from Riemann–Roch analysis, and formulates the Singleton-type defect. | | 8. Describes the isotrivial cover construction with \(Y=\mathrm{Spec}(\ell)\times_{\mathrm{Spec}(k)}X\), its unramified property, genus relation, and the specialization to linearized Reed–Solomon codes attaining MSRD. | | 9. Establishes the asymptotic performance framework: towers of curves with many rational points, construction of code families \((C_n)\), definitions of rate and relative distance, derivation of asymptotic bounds, and comparison with the sum-rank Gilbert–Varshamov bound. |