New Families of Optimal Codes

Updated 25 January 2026

The paper introduces a new family of optimal codes via explicit algebraic constructions that meet classical bounds like the Griesmer and Singleton bounds.
These constructions employ methods such as quasi-twisted codes, simplicial complexes, and trace-ring evaluations to achieve few-weight and distance-optimal properties.
The codes are designed for practical applications in communication, storage, and cryptography, offering clear algebraic transparency and efficient error correction.

A new family of optimal codes refers to an explicit construction or algebraic parameterization yielding linear codes—typically over finite fields or rings—with provably best-possible properties in terms of minimum distance, length, dimension, and related metrics. Recent development in this area has produced infinite parameterized families that meet classic benchmarks (such as the Griesmer bound, Singleton bound, and sphere-packing bound), often yielding optimal or distance-optimal codes with few-weights, projectivity, or prescribed locality. These codes underpin practical communication, storage, and authentication protocols and are characterized by their algebraic transparency and closed-form enumerators.

1. Algebraic Families: Quasi-Twisted and Quasi-Cyclic Two-Weight Codes

A major algebraic approach involves explicit constructions based on cyclic and consta-cyclic simplex codes, yielding infinite families of two-weight codes in quasi-twisted (QT) or quasi-cyclic (QC) formats. The construction works as follows (0712.0541):

For $q$ a prime power and $t>1$ , the $q$ -ary simplex code, dual to the Hamming code, has parameters $[m, t, q^{t-1}]_q$ with $m=(q^t-1)/(q-1)$ , and is equidistant (all nonzero codewords have weight $q^{t-1}$ ).
The code is realized as a $\lambda$ -consta-cyclic code via $g(x) = (x^m - \lambda)/h(x)$ where $h(x)$ is primitive of degree $t$ , and $\lambda\in\mathbb{F}_q^\times$ has order $q-1$ .
The generator matrix of the two-weight code is built as a $2\times p$ block matrix of $m\times m$ twistulant matrices, yielding a parameterized family $[pm, 2t; (p-1)q^{t-1}, p q^{t-1}]_q$ for $2\leq p\leq q^t$ .
Choice of $p$ enables explicit control of weights and allows for exact matches to the Griesmer bound in subcases such as $p = q^t-q+r+1$ for $r=1,\ldots,q$ .

Representative examples include binary QC codes $[195,8,96]$ , $[210,8,104]$ , $[240,8,120]$ , and ternary codes $[208,6,135]_3$ , $[221,6,144]_3$ ; the weight spectrum and optimality criteria (distance, length) are given in rigorous closed forms (0712.0541).

2. Geometric and Combinatorial Families: Affine/Projective Solomon-Stiffler Codes and Simplicial Complexes

Affine and projective Solomon-Stiffler code constructions generalize the classical approach by employing deletion of columns corresponding to lower-dimensional subspaces in vector spaces or projective geometries (Chen, 2024, Hu et al., 2023, Chen et al., 2024):

For $q$ prime power, $k\geq 3$ , columns corresponding to nonzero vectors in $S_1,\ldots,S_h\subseteq\mathbb{F}_q^k$ (pairwise disjoint, $\dim S_i=u_i$ ) are deleted, yielding an affine code $[n, k, d]_q$ with $n = q^k-1 - \sum_{i=1}^h(q^{u_i}-1)$ and $d \geq (q-1)(q^{k-1}- \sum_i q^{u_i-1})$ .
Modified versions employ deletion of orbits under multiplicative subgroups $E\subset\mathbb{F}_q^*$ , producing a projective code when $e=1$ .
Attainment of the Griesmer bound is shown to follow from intersection patterns of hyperplanes with deleted subspaces; defects can be exactly calculated, and optimal (zero-defect) families are produced for suitable choices of parameters.
Simplicial complex techniques generalize this by encoding support structures and inclusion-exclusion properties into the defining set $D$ , recovering all classical projective few-weight codes, and enabling explicit control of Griesmer, near-Griesmer, or distance-optimality (Hu et al., 2023).

Families feature explicit two-weight, three-weight, up to five-weight distributions, and precise connection of combinatorial support patterns to code metrics.

3. Trace-Ring and Ring-Based Constructions: Few-Weight and Two-Weight Optimal Codes

Trace constructions over semi-local rings $\mathbb{F}_q+u\mathbb{F}_q$ (with $u^2=0$ ) have led to several new infinite families of codes with precisely determined Lee or Hamming weight distributions and optimality properties (Shi et al., 2016, Wu et al., 2019, Chen et al., 2024):

Codes are constructed via trace-evaluation over sets structured by abelian group actions (e.g., squares, union of subfields, coset representatives, or combinatorial designs).
The Gray map is exploited to translate Lee-wise optimal codes over rings into distance- or Griesmer-optimal codes over finite fields.
Two-weight codes with parameters $[(p^m-1)^2,2m,d]$ and $[2(p^m-1)^2,2m,d']$ are proven to meet the Griesmer bound (under arithmetic conditions $p\equiv 3\pmod{4}$ , $m$ odd, etc.), and Lee weight enumerators are computed via Gauss sum and character-theoretic evaluations (Shi et al., 2016).
Simplicial complex ring constructions generalize these further to arbitrary prime powers, producing codes with 4–10 weights and multiple infinite families of near-Griesmer and distance-optimal Gray images (Chen et al., 2024, Wu et al., 2019).

4. Classical and Block Design Approaches: Constant-Weight Codes, Combinatorial Batch Codes

Several combinatorial design frameworks yield new optimal constant-weight and batch codes (0705.0081, Silberstein et al., 2013):

By “lifting” $s$ disjoint optimal binary constant-weight codes to $q$ -ary via re-coloring, and employing packings or Steiner systems (STS), one constructs infinite families that match Johnson and Svanström bounds and are provably optimal.
For batch codes, block design constructions (affine planes, transversal designs) produce optimal $c$ -uniform codes where $c \sim \sqrt{k}$ , filling previously unknown parameter regimes. Hall condition guarantees retrieval property and explicit optimality is checked via combinatorial incidence (Silberstein et al., 2013).
These approaches show combinatorial methods can yield optimal codes in scenarios not easily accessible by algebraic means.

5. Locally Repairable Codes (LRCs), Self-Dual Codes, and Rank-Metric Codes

Explicit algebraic and matrix-product constructions have produced new infinite families of optimal LRCs and self-dual codes:

Anticode deletion, subspace codes, and matrix-product techniques generate binary and $q$ -ary codes meeting LRC Singleton-type and Griesmer bounds, extending locality to arbitrary small parameters (Silberstein et al., 2015, Andrade et al., 2021, Luo et al., 2021).
Algebraic geometry codes on curves (projective line, elliptic curves, Hermitian curves) produce optimal self-dual (MDS and almost-MDS) codes for previously unattainable lengths and field sizes (Sok, 2020, Sok, 2021).
New symmetric, alternating, and Hermitian rank-metric codes with reversible encoding and interpolation-based decoding achieve optimal parameters and unique error correction up to half the minimum rank distance (Kadir et al., 2022).

6. Bounds and Weight Distribution: Griesmer, Singleton, Sphere-Packing

A persistent feature of all these families is explicit analysis of code optimality with respect to classic bounds:

The Griesmer bound is central in determining length-optimality; codes are often constructed to meet this bound exactly or with defect 1 (near-Griesmer); explicit formulae for $n$ vs. $k, d$ facilitate systematic verification (Chen, 2024, 0712.0541, Chen et al., 2024, Hu et al., 2023).
Sphere-packing, Singleton-type, and various code-specific bounds (e.g., Cadambe-Mazumdar for LRCs, SR-optimality for AMD codes) drive the selection of parameters and constructions (Huczynska et al., 2018, Silberstein et al., 2015).
Weight distributions are computed in closed form using intersection, combinatorial, or character theory, yielding linear codes with precisely determined few-weight or two-weight spectra, critical for applications in secret sharing, authentication, and quantum coding (Shi et al., 2016, Chen, 2024, Chen et al., 2024, Silberstein et al., 2013).

7. Impact, Applications, and Landscape

These new families have substantial impact:

They provide large tables of codes with optimal parameters (distance, length), improve flexibility (variable locality, weight, hull-dimension), and unify prior ad hoc or sporadic constructions under algebraic or geometric frameworks.
Their explicitness allows for rigorous proofs of optimality, transparent weight spectrum, and immediate utility in implementation for error correction, storage, authentication, cryptography, and quantum coding.
The unification of algebraic, combinatorial, geometric, and ring-based constructions presents a comprehensive landscape for optimal codes, where researchers can systematically access infinite or parameterized families meeting the most stringent theoretical bounds across applications.

Summary Table: Representative Infinite Families of New Optimal Codes

Construction Type	Example Parameters	Optimality
2-generator QT/QC	$[195,8,96]$ , $[240,8,120]$	Griesmer, distance
Affine Solomon-Stiffler	$[64,4,42]_3$ , $[208,5,138]_3$	Griesmer, few-weight
Trace-ring simplicial	$[2^{2m-1},2m,2^{2m-2}]_2$	Griesmer, two-weight
Constant-weight/des. batch	$[n, w, d]_q$ via STS, affine	Johnson, optimal
LRC matrix-product	$[n, k, d]_q$ not divisible	Singleton, optimal
AG self-dual MDS	$[26,13,14]$ , $[30,9,18]_{25}$	MDS, hull-optimal
Rank-metric (symmetric)	$n(n-d+2)/2$	Singleton, optimal