Self-Orthogonal Minimal Linear Codes

• Self-Orthogonal Minimal Linear Codes are codes that embed into a larger self-orthogonal structure with minimal extra coordinates while ensuring every nonzero codeword is support-minimal.
• They employ diverse methods—ranging from generator-matrix parity checks to function-based constructions—to yield explicit parameters and optimal minimum distances in low dimensions.
• These codes are pivotal for applications in quantum error correction and secure secret sharing, while also presenting open challenges in higher-dimensional and nonbinary settings.

Searching arXiv for papers on self-orthogonal minimal linear codes and related embedding results. Self-orthogonal minimal linear codes occupy the intersection of two active strands of coding theory. In one strand, a code is made self-orthogonal by embedding it into a larger code with the fewest possible added coordinates, so “minimal” refers to shortest-length self-orthogonal embedding (Kim et al., 2020). In the other, “minimal” is used in the support-theoretic sense: every nonzero codeword is minimal under support inclusion, often in families that also satisfy self-orthogonality and may violate the Ashikhmin–Barg condition (Li et al., 16 Jul 2025). For binary codes, self-orthogonality means $C \subseteq C^\perp$, equivalently $G G^{\mathsf T}=0$ for a generator matrix $G$; it forces every codeword to have even weight and implies the dimension bound $k \le n/2$ (Kim et al., 2020).

1. Terminology and foundational constraints

The literature uses “minimal” in two non-equivalent senses, and distinguishing them is necessary for reading recent work correctly.

Notion	Meaning	Representative papers
Minimal embedding	Shortest self-orthogonal extension of a given code	(Kim et al., 2020, An et al., 7 Nov 2025, An et al., 8 Jun 2026)
Minimal code	Every nonzero codeword is support-minimal	(Li et al., 16 Jul 2025, Chen et al., 11 May 2025, Jin et al., 14 Jul 2025)

For binary linear codes, a code $C \subseteq \mathbb{F}_2^n$ is self-orthogonal if $C \subseteq C^\perp$, or equivalently if $G G^{\mathsf T}=0$. This implies $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$ for every $x \in C$, so binary self-orthogonal codes are even, and their minimum distance is even. Doubly-even binary codes are automatically self-orthogonal, while singly-even self-orthogonal codes require finer criteria (Kim et al., 2020, Li et al., 16 Jul 2025).

Support-minimality is defined through the support order. A nonzero codeword $c$ is minimal if every codeword with support contained in $G G^{\mathsf T}=0$0 is a nonzero scalar multiple of $G G^{\mathsf T}=0$1. The Ashikhmin–Barg condition gives a sufficient criterion: for a $G G^{\mathsf T}=0$2-ary linear code with minimum and maximum nonzero weights $G G^{\mathsf T}=0$3 and $G G^{\mathsf T}=0$4,

$G G^{\mathsf T}=0$5

implies minimality. Much recent work on self-orthogonal minimal codes is concerned precisely with constructing families that remain minimal even when this inequality fails (Li et al., 16 Jul 2025, Chen et al., 11 May 2025).

2. Characterizations of self-orthogonality

A central binary characterization states that a binary linear code $G G^{\mathsf T}=0$6 is self-orthogonal if and only if

$G G^{\mathsf T}=0$7

This reformulates orthogonality in terms of weight congruences and immediately yields the standard facts that self-orthogonal binary codes are even and doubly-even codes are automatically self-orthogonal (Li et al., 16 Jul 2025).

The same paper gives a precise criterion for the singly-even case. If

$G G^{\mathsf T}=0$8

then $G G^{\mathsf T}=0$9 is self-orthogonal and singly-even if and only if $G$0 is a binary $G$1 linear code. This identifies singly-even self-orthogonal codes as one-coset extensions of a doubly-even subcode of codimension $G$2 (Li et al., 16 Jul 2025).

A different, generator-matrix-based line of work characterizes binary self-orthogonality through column multiplicities. Let $G$3 be the generator matrix of the binary simplex code, with nonzero columns $G$4, and let $G$5 count occurrences of $G$6 among the columns of a $G$7 generator matrix $G$8. Then a general criterion states that $G$9 is self-orthogonal if and only if

$k \le n/2$0

where $k \le n/2$1 indexes columns having $k \le n/2$2 in the $k \le n/2$3-th relevant position. For $k \le n/2$4, this reduces to

$k \le n/2$5

and for $k \le n/2$6 to

$k \le n/2$7

For $k \le n/2$8, self-orthogonality is described by parity congruences on two explicit index partitions $k \le n/2$9 and $C \subseteq \mathbb{F}_2^n$0, $C \subseteq \mathbb{F}_2^n$1 (Kim et al., 2020).

For odd prime alphabets, recent function-based work uses different criteria. Over $C \subseteq \mathbb{F}_2^n$2, a linear code is self-orthogonal if and only if every codeword has weight divisible by $C \subseteq \mathbb{F}_2^n$3. More generally, for odd $C \subseteq \mathbb{F}_2^n$4, a code is self-orthogonal if and only if $C \subseteq \mathbb{F}_2^n$5 for all $C \subseteq \mathbb{F}_2^n$6. In constructions from $C \subseteq \mathbb{F}_2^n$7-ary functions, this can be expressed in Walsh-expansion form: if $C \subseteq \mathbb{F}_2^n$8, then $C \subseteq \mathbb{F}_2^n$9 is self-orthogonal if and only if $C \subseteq C^\perp$0 for all $C \subseteq C^\perp$1 (Jin et al., 14 Jul 2025).

3. Minimality as shortest self-orthogonal embedding

The binary embedding problem asks for the minimum number of coordinates that must be appended to a generator matrix $C \subseteq C^\perp$2 so that the resulting code is self-orthogonal while preserving the original dimension. For dimensions $C \subseteq C^\perp$3, explicit shortest-length algorithms were given using the column-parity characterizations. For $C \subseteq C^\perp$4, one appends a column $C \subseteq C^\perp$5 whenever the corresponding multiplicity $C \subseteq C^\perp$6 is odd, adding at most three columns. For $C \subseteq C^\perp$7, one iteratively equalizes parity across the seven nonzero column types, adding exactly $C \subseteq C^\perp$8 columns. For $C \subseteq C^\perp$9, one minimizes over the fifteen parity-partition systems and adds at most five columns. For $G G^{\mathsf T}=0$0, a recursive reduction to the $G G^{\mathsf T}=0$1-dimensional case was proposed; for $G G^{\mathsf T}=0$2, the stated upper bound is $G G^{\mathsf T}=0$3 added columns. In all of these constructions, the embedded code has the same dimension and minimum distance $G G^{\mathsf T}=0$4 (Kim et al., 2020).

The same work derives exact optimal minimum distances for self-orthogonal codes in low dimension. It gives new explicit formulas for $G G^{\mathsf T}=0$5 for all $G G^{\mathsf T}=0$6, resolving the cases $G G^{\mathsf T}=0$7 left open by Li–Xu–Zhao, and it gives formulas for $G G^{\mathsf T}=0$8 whenever $G G^{\mathsf T}=0$9, together with a conjectural description of the excluded residue classes. MAGMA computations reported there indicate that the embedding procedures send optimal linear codes to optimal self-orthogonal codes in the tested ranges (Kim et al., 2020).

A later hull-theoretic treatment determines the shortest embedding length of any binary linear code in terms of the hull $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$0. If $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$1, then the minimum number $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$2 of added columns is controlled by $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$3: if $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$4 is odd, then $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$5; if $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$6 is even and $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$7 is even, then $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$8. This yields an exact length formula $x \cdot x = \mathrm{wt}(x) \equiv 0 \pmod{2}$9 and shows, for example, that every shortest self-orthogonal embedding of the binary Hamming code $x \in C$0 is self-dual. The same framework constructs a self-dual $x \in C$1 code from $x \in C$2, identified there as the shortened Golay code, and a self-dual $x \in C$3 code from $x \in C$4 (An et al., 7 Nov 2025).

The embedding problem has also been extended to $x \in C$5-linear codes. There the minimal length is measured by the shortest self-orthogonal embedding over $x \in C$6, and tight bounds are given in terms of the type $x \in C$7 of the Gram-module generated by $x \in C$8: $x \in C$9 A residue-code criterion relates this to the shortest doubly-even self-orthogonal embedding of $c$0, and exact lengths are obtained for quaternary Preparata codes (An et al., 8 Jun 2026).

4. Minimality as support minimality and the Ashikhmin–Barg frontier

In the support-theoretic sense, a binary linear code $c$1 is minimal if every nonzero codeword is minimal in the support order. One criterion used in this literature is that $c$2 is minimal if and only if, for any distinct nonzero codewords $c$3,

$c$4

For binary two-weight codes with nonzero weights $c$5 and $c$6, minimality follows immediately (Li et al., 16 Jul 2025).

A recurrent theme is the construction of minimal codes that violate the Ashikhmin–Barg condition. One general method begins from a minimal code satisfying Ashikhmin–Barg and appends

$c$7

coordinates that are nonzero only in the first generator row. The resulting code remains minimal, preserves $c$8, raises the maximum weight to at least $c$9, and therefore forces

$G G^{\mathsf T}=0$00

This produces minimal codes violating Ashikhmin–Barg from arbitrary projective linear codes after a simplex-complementary intermediate construction, and in the binary doubly-even setting it yields self-orthogonal families as well (Chen et al., 11 May 2025).

Recent work on binary singly-even self-orthogonal codes goes further by constructing families that are simultaneously self-orthogonal, singly-even, minimal, and Ashikhmin–Barg-violating. In these constructions, minimality is established either through the two-weight criterion or via puncturing arguments that transfer minimality from a known minimal source code. The resulting families include few-weight codes derived from simplex codes, bent-function codes, and Boolean-function constructions, all with explicit weight distributions (Li et al., 16 Jul 2025).

A complementary function-theoretic paper states that, to the authors’ knowledge, it is the first to investigate constructions of linear codes that violate the Ashikhmin–Barg condition and satisfy self-orthogonality. It produces binary and $G G^{\mathsf T}=0$01-ary self-orthogonal minimal families from Boolean, quadratic-trace, and plateaued-function constructions, with exact weight distributions and explicit self-orthogonality criteria in Walsh-transform form (Jin et al., 14 Jul 2025).

Not every self-orthogonal few-weight construction is known to be minimal. Codes built from weakly regular plateaued functions are shown to be self-orthogonal and optimally or almost optimally extendable, but the same paper explicitly notes that the Ashikhmin–Barg sufficient condition fails for those families and that minimality is not established there. This serves as a useful correction to the common conflation of self-orthogonality, few-weight structure, and minimality (Wang et al., 2024).

5. Construction paradigms

A general binary construction from known self-orthogonal codes proceeds by block concatenation. If $G G^{\mathsf T}=0$02 is an $G G^{\mathsf T}=0$03 self-orthogonal code with generator rows $G G^{\mathsf T}=0$04, and $G G^{\mathsf T}=0$05 is an $G G^{\mathsf T}=0$06 self-orthogonal code, then choosing arbitrary $G G^{\mathsf T}=0$07 and forming

$G G^{\mathsf T}=0$08

produces an $G G^{\mathsf T}=0$09 self-orthogonal code. From this template come three infinite classes of binary self-orthogonal singly-even codes with few weights: a two-weight family from simplex codes, a four-weight family from simplex codes with two appended patterns, and a four-weight family obtained from bent-function codes. The same paper also gives Boolean-function criteria: for a code $G G^{\mathsf T}=0$10 derived from a Boolean function $G G^{\mathsf T}=0$11, $G G^{\mathsf T}=0$12 is self-orthogonal if and only if $G G^{\mathsf T}=0$13 for all $G G^{\mathsf T}=0$14, and $G G^{\mathsf T}=0$15 is self-orthogonal singly-even if and only if $G G^{\mathsf T}=0$16 for all $G G^{\mathsf T}=0$17 (Li et al., 16 Jul 2025).

Function-based constructions over $G G^{\mathsf T}=0$18 and odd prime fields broaden this picture. One binary family arises from products of quadratic trace forms and yields doubly-even self-orthogonal minimal codes violating Ashikhmin–Barg. Another binary family is singly-even, constructed from a product together with two puncture terms. For odd $G G^{\mathsf T}=0$19, two additional classes come from $G G^{\mathsf T}=0$20-ary functions: one from the special function $G G^{\mathsf T}=0$21, giving self-orthogonal two-weight codes that are optimal in the case $G G^{\mathsf T}=0$22, and one from unbalanced weakly regular plateaued functions, producing self-orthogonal minimal codes with four to six nonzero weights, depending on $G G^{\mathsf T}=0$23 and the parity of $G G^{\mathsf T}=0$24 (Jin et al., 14 Jul 2025).

Ring-based Gray-image methods provide another major source of self-orthogonal minimal binary codes. Over the non-unital ring $G G^{\mathsf T}=0$25 of size $G G^{\mathsf T}=0$26, $G G^{\mathsf T}=0$27-type constructions produce few-weight binary Gray images that are self-orthogonal when $G G^{\mathsf T}=0$28; under mild Ashikhmin–Barg-type inequalities, these Gray images are minimal, and an infinite class is distance-optimal with respect to the Griesmer bound (Sagar et al., 2023). Over the mixed alphabet ring $G G^{\mathsf T}=0$29, the Gray image $G G^{\mathsf T}=0$30 is self-orthogonal for $G G^{\mathsf T}=0$31, and when $G G^{\mathsf T}=0$32 with $G G^{\mathsf T}=0$33, $G G^{\mathsf T}=0$34 is minimal by an explicit Ashikhmin–Barg computation (Kewat et al., 2022). Over the non-chain ring $G G^{\mathsf T}=0$35, the Gray image $G G^{\mathsf T}=0$36 is self-orthogonal when $G G^{\mathsf T}=0$37 are nonempty, while associated binary subfield-like codes $G G^{\mathsf T}=0$38 include one-weight and two-weight minimal families and many Griesmer-optimal instances (Sagar et al., 2022).

6. Parameters, applications, and open problems

The recent literature contains a substantial amount of exact parameter determination. For binary self-orthogonal codes, exact formulas for optimal minimum distances are known for all $G G^{\mathsf T}=0$39 self-orthogonal codes and for many $G G^{\mathsf T}=0$40 self-orthogonal codes (Kim et al., 2020). Hull-based shortest-embedding methods have also produced many inequivalent optimal self-orthogonal codes in dimensions $G G^{\mathsf T}=0$41 and $G G^{\mathsf T}=0$42, including the parameter sets $G G^{\mathsf T}=0$43, $G G^{\mathsf T}=0$44, $G G^{\mathsf T}=0$45, and $G G^{\mathsf T}=0$46 (An et al., 7 Nov 2025). In the $G G^{\mathsf T}=0$47 setting, a shortest-embedding algorithm found twelve linear codes whose minimum Lee distances exceed those of the $G G^{\mathsf T}=0$48-linear codes in Aydin’s database (An et al., 8 Jun 2026).

Applications recur across these works. Self-orthogonal codes are repeatedly linked to CSS quantum-code constructions, and in some papers also to lattice design and related areas (Li et al., 16 Jul 2025, Wang et al., 2024). Minimal codes are tied to secret sharing and secure two-party computation, because support-minimal codewords determine access structures in linear secret-sharing schemes (Chen et al., 11 May 2025, Jin et al., 14 Jul 2025). This suggests that the intersection of self-orthogonality and minimality is especially valuable when both dual-containing structure and support-theoretic rigidity are required simultaneously.

Several open directions remain explicit in the current record. For binary embeddings, extending shortest-length guarantees beyond the recursive methods known for $G G^{\mathsf T}=0$49 remains open, as does the classification of optimal self-orthogonal codes in higher dimensions (Kim et al., 2020). For support-minimal codes, open problems include nonbinary generalizations, tighter analyses of dual distances and automorphism groups, and systematic classification of Boolean functions satisfying the congruence $G G^{\mathsf T}=0$50 in the singly-even self-orthogonal setting (Chen et al., 11 May 2025, Li et al., 16 Jul 2025). For $G G^{\mathsf T}=0$51-linear codes, general computation of the invariant $G G^{\mathsf T}=0$52 and systematic optimization of Lee distance under shortest self-orthogonal embeddings are singled out as unresolved (An et al., 8 Jun 2026). Finally, in the plateaued-function literature, minimality itself remains unproved for some self-orthogonal families, indicating that self-orthogonality, few-weight structure, optimal extendability, and minimality still form a partially rather than fully unified theory (Wang et al., 2024).