Griesmer Optimal Linear Codes Overview

• Griesmer optimal linear codes are defined by meeting the sum-of-ceilings bound, achieving the minimum possible length for given parameters.
• They incorporate constructions such as simplex and Solomon–Stiffler methods, alongside dual anticode and b-symbol metric extensions.
• Their rigid combinatorial structure and recursive subcode properties offer robust error correction and applications in design theory.

Searching arXiv for recent and foundational papers on Griesmer-optimal linear codes and closely related constructions. Querying arXiv for “Griesmer optimal linear codes”, “Solomon-Stiffler”, “few-weight”, and “b-symbol Griesmer”. Griesmer optimal linear codes are linear codes whose length attains the Griesmer lower bound for the prescribed alphabet size, dimension, and minimum distance. For a $q$-ary $[n,k,d]_q$ linear code, the classical bound is

$$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$

and a code with $n=g_q(k,d)$ is a Griesmer code in the classical sense (Chen et al., 2024). Recent work has also formalized the Griesmer defect $g(C)=n-g_q(k,d)$, so $g(C)=0$ means Griesmer optimal and $g(C)=1$ means almost Griesmer in that convention (Chen et al., 2024). Closely related notions include near Griesmer codes with $n-1=g_q(k,d)$ (Hu et al., 2023), and extensions of the Griesmer idea to generalized Hamming weights, the $b$-symbol metric, and additive codes (Pan et al., 2024, Luo et al., 2024, Kurz, 2024).

1. Classical bound, defects, and terminology

The classical Griesmer bound is a length lower bound for Hamming-metric linear codes. It is one of the standard benchmarks for optimality because meeting it forces the length to be as small as possible for the given $(q,k,d)$, hence implies distance-optimality in the usual coding-theoretic sense (Chen et al., 2024, Chen, 2024). In a recent formalization, the defect

$[n,k,d]_q$0

measures the gap to the bound; this makes it convenient to distinguish Griesmer codes ($[n,k,d]_q$1) from almost Griesmer codes ($[n,k,d]_q$2) (Chen et al., 2024).

Terminology is not completely uniform across the literature. One line of work uses almost Griesmer for defect one (Chen et al., 2024), while another uses near Griesmer for $[n,k,d]_q$3 (Hu et al., 2023). These notions agree in spirit but are not identical. Recent work on generalized Hamming weights introduces a further refinement: for an $[n,k,d]_q$4 code $[n,k,d]_q$5, the $[n,k,d]_q$6-th generalized Hamming weight $[n,k,d]_q$7 satisfies

$[n,k,d]_q$8

and the corresponding $[n,k,d]_q$9-Griesmer defect

$$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$0

allows the definition of an $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$1-Griesmer code as one with $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$2 at the smallest possible $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$3 (Pan et al., 2024).

A recurrent misconception is to identify “optimal” with “Griesmer optimal.” Recent constructions repeatedly separate strict Griesmer codes from near-Griesmer and merely distance-optimal families. Affine Solomon–Stiffler constructions produce examples with defect $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$4, $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$5, or $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$6, all still optimal in the sense that no larger minimum distance exists for the same $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$7 (Chen, 2024). Simplicial-complex constructions likewise yield Griesmer, near-Griesmer, and distance-optimal families under explicit combinatorial criteria (Hu et al., 2023).

2. Canonical families and the Solomon–Stiffler paradigm

The simplex code is the archetypal Griesmer-optimal linear code. It has parameters

$$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$8

and is a one-weight code; it meets the classical Griesmer bound and, more strongly, meets all generalized Griesmer bounds for $$n \ge g_q(k,d):=\sum_{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil,$$9 (Pan et al., 2024). Every $n=g_q(k,d)$0-dimensional subcode of a simplex code has support weight

$n=g_q(k,d)$1

so its subcode-support-weight distributions are completely rigid (Pan et al., 2024).

A second foundational paradigm is the Solomon–Stiffler construction. In its original binary projective form, one starts from the binary simplex $n=g_q(k,d)$2 code and deletes columns lying in prescribed subspaces $n=g_q(k,d)$3. The resulting code has parameters

$n=g_q(k,d)$4

and meets the Griesmer bound (Chen, 2024). Recent work recasts this as part of a broader affine Solomon–Stiffler framework, and then a modified affine Solomon–Stiffler framework obtained by thinning scalar multiples by a subgroup of $n=g_q(k,d)$5 (Chen, 2024).

Several representative families produced by these constructions are explicit Griesmer codes.

Construction	Parameters	Status
Simplex	$n=g_q(k,d)$6	Griesmer; all generalized Griesmer bounds met
Modified affine Solomon–Stiffler	$n=g_q(k,d)$7	Griesmer if $n=g_q(k,d)$8 (Chen, 2024)
Subfield-complement code	$n=g_q(k,d)$9	Griesmer (Hu et al., 2021)
GRS complement in simplex	$g(C)=n-g_q(k,d)$0	1-Griesmer if $g(C)=n-g_q(k,d)$1; 2-Griesmer if $g(C)=n-g_q(k,d)$2 (Pan et al., 2024)
Binary Gray-image family	$g(C)=n-g_q(k,d)$3	Meets Griesmer bound (Wu et al., 2019, Xu et al., 2022)

This landscape is broader than the classical projective picture. A subfield-based construction from complements of unions of subfields or of cosets of a subfield yields infinite Griesmer and near-Griesmer families with explicit criteria from the Griesmer bound; for example, $g(C)=n-g_q(k,d)$4 is Griesmer, while $g(C)=n-g_q(k,d)$5 is near-Griesmer in the multiplicative-coset family (Hu et al., 2021). Simplicial-complex constructions over $g(C)=n-g_q(k,d)$6 produce projective codes whose Griesmer status is controlled by the support sets $g(C)=n-g_q(k,d)$7: pairwise disjoint supports and bounded multiplicity of equal dimensions are necessary and sufficient for a Griesmer code, thereby recovering Solomon–Stiffler codes in that language (Hu et al., 2023).

3. Duality with anticodes and complementary constructions

A recent conceptual shift is to study Griesmer optimality through projective linear anticodes. A $g(C)=n-g_q(k,d)$8-ary anticode $g(C)=n-g_q(k,d)$9 has diameter

$g(C)=0$0

so distances are bounded above rather than below. For a projective linear code $g(C)=0$1 of dimension $g(C)=0$2 with maximum weight $g(C)=0$3, and assuming $g(C)=0$4, the antiGriesmer bound is

$g(C)=0$5

The associated antiGriesmer defect is

$g(C)=0$6

and $g(C)=0$7 defines an antiGriesmer code (Chen et al., 2024).

The importance of this bound is that it is exactly dual to the classical Griesmer bound under the complementary construction inside the simplex code. If $g(C)=0$8 is a projective linear code of dimension $g(C)=0$9 and length $g(C)=1$0, deleting its columns from a simplex generator matrix produces a complementary code $g(C)=1$1 with parameters

$g(C)=1$2

From the proof of the antiGriesmer theorem, the complementary code of a projective Griesmer code is antiGriesmer, and conversely (Chen et al., 2024). This gives a systematic mechanism for constructing Griesmer or almost Griesmer few-weight codes from anticodes with small antiGriesmer defect.

This complementary viewpoint yields explicit classical families. The complementary MDS code

$g(C)=1$3

is a minimal $g(C)=1$4-weight Griesmer code when $g(C)=1$5, and becomes almost Griesmer when $g(C)=1$6 (Chen et al., 2024). The complementary Reed–Solomon code

$g(C)=1$7

is a minimal $g(C)=1$8-weight almost Griesmer code (Chen et al., 2024). A notable corollary is that for $g(C)=1$9 one obtains a minimal three-weight Griesmer code

$n-1=g_q(k,d)$0

(Chen et al., 2024).

A further point of contrast with unrestricted combinatorics is that the antiGriesmer bound is strictly stronger than the Erdős–Kleitman bound when restricted to binary projective linear anticodes. In particular, although unrestricted binary anticodes of diameter $n-1=g_q(k,d)$1 may have arbitrarily large length, the antiGriesmer machinery implies that a binary projective linear anticode of diameter $n-1=g_q(k,d)$2 and $n-1=g_q(k,d)$3 must satisfy $n-1=g_q(k,d)$4 (Chen et al., 2024).

4. Generalized, metric, and additive extensions

The concept of Griesmer optimality now extends well beyond the classical Hamming metric. The most systematic extension is via generalized Hamming weights. The simplex code remains the reference object because it meets the generalized Griesmer bound for every $n-1=g_q(k,d)$5, but recent work constructs codes that are $n-1=g_q(k,d)$6-Griesmer for a specified $n-1=g_q(k,d)$7 without being 1-Griesmer. In one family,

$n-1=g_q(k,d)$8

the resulting code is $n-1=g_q(k,d)$9-Griesmer, almost Griesmer at level $b$0, and distance-optimal (Pan et al., 2024). A modified Solomon–Stiffler family with

$b$1

has the same qualitative behavior: it is $b$2-Griesmer, almost Griesmer, and distance-optimal (Pan et al., 2024).

A second extension replaces the Hamming metric by the $b$3-symbol metric. For an $b$4 linear code, the $b$5-symbol Griesmer bound is

$b$6

equivalently

$b$7

This reduces to the classical Griesmer bound when $b$8 (Luo et al., 2024). Two explicit distance-optimal families are

$b$9

for cyclic codes and

$(q,k,d)$0

for extended cyclic codes; both attain the $(q,k,d)$1-symbol Griesmer bound (Luo et al., 2024). A later paper determines the optimal parameters of linear codes in the $(q,k,d)$2-symbol metric when the minimum distance is sufficiently large, and determines the optimal parameters of linear binary codes in the pair-symbol metric for small dimensions (Kurz, 10 Jul 2025).

A third extension is to additive codes over $(q,k,d)$3. If an additive code is viewed as an $(q,k,d)$4-linear object of rank $(q,k,d)$5, the Griesmer-type inequality becomes

$(q,k,d)$6

and recent work proves that for every fixed $(q,k,d)$7 and $(q,k,d)$8 there exists $(q,k,d)$9 such that for all $[n,k,d]_q$00, the maximum length $[n,k,d]_q$01 attains this upper bound exactly (Kurz, 2024). That work also exhibits many infinite families where additive codes outperform the corresponding linear constructions, showing that classical linear Griesmer optimality is not globally optimal once additive structures are admitted (Kurz, 2024).

5. Arithmetic, geometric, and asymptotic structure

Griesmer optimality imposes strong internal rigidity. A 2025 divisibility theorem shows that if $[n,k,d]_q$02 is a Griesmer code $[n,k,d]_q$03 over $[n,k,d]_q$04 with $[n,k,d]_q$05 and $[n,k,d]_q$06, then every codeword weight is divisible by $[n,k,d]_q$07: $[n,k,d]_q$08 More generally, if $[n,k,d]_q$09, then every weight is divisible by

$[n,k,d]_q$10

(Deng et al., 9 Jun 2025). This extends Ward’s prime-field divisibility theorem and clarifies a common misconception: over extension fields the correct divisibility conclusion is by a power of $[n,k,d]_q$11, not necessarily by the same power of $[n,k,d]_q$12; the quaternary hexacode is the standard counterexample to the stronger statement (Deng et al., 9 Jun 2025).

The same paper proves a basis theorem: if $[n,k,d]_q$13, then a Griesmer code admits a basis $[n,k,d]_q$14 such that the first $[n,k,d]_q$15 vectors span a constant-weight Griesmer subcode of parameters $[n,k,d]_q$16, and any $[n,k,d]_q$17 basis vectors span a $[n,k,d]_q$18 Griesmer subcode (Deng et al., 9 Jun 2025). This suggests that Griesmer optimal codes are recursively saturated by smaller Griesmer optimal faces, much as MDS codes are stable under shortening.

Projective geometry supplies another invariant. For a Griesmer code $[n,k,d]_q$19, the maximum projective point multiplicity satisfies

$[n,k,d]_q$20

and $[n,k,d]_q$21 is projective iff $[n,k,d]_q$22 (Deng et al., 9 Jun 2025). In the binary case this interacts with minihyper theory and with Solomon–Stiffler or Belov-type geometries.

A further structural parameter is the error coefficient $[n,k,d]_q$23, the number of minimum-weight codewords. For binary Griesmer optimal codes, recent work derives five iterative lower bounds on $[n,k,d]_q$24 by combining residual-code recursion, Griesmer subcode chains, and minihyper rank arguments. These bounds are tight in most cases when the dimension is at most $[n,k,d]_q$25, and even when not tight their gap to the actual value is at most $[n,k,d]_q$26 (Guan et al., 8 Jul 2025). In the AWGN setting under maximum-likelihood decoding, minimizing $[n,k,d]_q$27 among optimal codes gives the best asymptotic frame error rate, leading to the notion of AFER-optimal Griesmer optimal codes (Guan et al., 8 Jul 2025).

6. Few-weight realizations and combinatorial consequences

A striking empirical regularity is that many Griesmer or near-Griesmer families are also few-weight codes. Affine and modified affine Solomon–Stiffler constructions produce explicit two-, three-, four-, and five-weight optimal families, and reconstruct many entries in Grassl’s tables as special cases (Chen, 2024). Simplicial-complex constructions over $[n,k,d]_q$28 yield at most $[n,k,d]_q$29-, $[n,k,d]_q$30-, and $[n,k,d]_q$31-weight families for one, two, and three maximal elements, together with necessary and sufficient conditions for being Griesmer or near-Griesmer (Hu et al., 2023). Trace-code constructions over $[n,k,d]_q$32 give two-weight Griesmer families

$[n,k,d]_q$33

when

$[n,k,d]_q$34

and many almost Griesmer examples beyond that regime (Jian, 2018). Subfield-code constructions produce a two-weight $[n,k,d]_q$35 family meeting the Griesmer bound, and the punctured binary family $[n,k,d]_q$36 also meeting it (Xu et al., 2022).

Gray-image constructions over $[n,k,d]_q$37 yield the same binary family $[n,k,d]_q$38 as an infinite Griesmer-optimal family arising from simplicial complexes (Wu et al., 2019). More generally, ring-based constructions over $[n,k,d]_q$39 and $[n,k,d]_q$40 produce free ring-linear codes attaining the ring Griesmer bound, and in the $[n,k,d]_q$41 case their Gray images give two-weight linear $[n,k,d]_q$42-codes that reach the classical Griesmer bound in suitable parameter ranges (Li et al., 2016). Recent affine Solomon–Stiffler analysis explicitly notes that several families previously obtained as Gray images of ring codes in 2017 and 2019 are subsumed by the affine or modified affine geometric framework, with identical parameters and weight distributions (Chen, 2024).

These families carry nontrivial combinatorial payload. Three-weight projective binary codes constructed from complementary anticodes lead to $[n,k,d]_q$43-strongly walk-regular graphs for every odd integer $[n,k,d]_q$44 (Chen et al., 2024). Subfield-code families produce $[n,k,d]_q$45- and $[n,k,d]_q$46-designs, including Steiner systems $[n,k,d]_q$47 from the $[n,k,d]_q$48 family (Xu et al., 2022). Few-weight trace codes and their Griesmer-optimal subclasses are minimal or self-orthogonal in many cases, which is relevant for secret sharing and quantum coding (Jian, 2018, Hu et al., 2021).

Taken together, these developments show that Griesmer optimal linear codes form a large and highly structured class rather than a collection of isolated extremal examples. The simplex code remains the universal model, but recent work has added dual anticode constructions, affine Solomon–Stiffler generalizations, simplicial-complex and subfield-based families, generalized-Hamming-weight analogues, $[n,k,d]_q$49-symbol variants, and arithmetic divisibility theorems. This suggests that the modern theory of Griesmer optimality is best understood as the interaction of extremal length–distance inequalities with projective geometry, subspace combinatorics, and recursive optimal subcode structure (Chen, 2024, Chen et al., 2024, Pan et al., 2024, Deng et al., 9 Jun 2025).