---
title: Doubly Even Binary Codes
url: https://www.emergentmind.com/topics/doubly-even-binary-codes
type: topic
---

# Doubly Even Binary Codes

A doubly even binary code is a linear subspace \(C \subseteq \mathbb{F}_2^n\) in which every codeword's Hamming weight is divisible by 4. These codes play a fundamental role in both coding theory and algebraic structures such as code loops. Doubly even self-dual codes—those which are equal to their dual and admit only weights divisible by 4—exist only for lengths \(n \equiv 0 \pmod{8}\) and satisfy strong combinatorial and algebraic constraints. The construction and classification of extremal doubly even self-dual codes, which attain the Mallows–Sloane bound on minimum distance, constitute a central theme in contemporary research.

## 1. Definition and Basic Properties

A binary linear code of length \(n\) is a subspace \(C \subseteq \mathbb{F}_2^n\). The Hamming weight \(\mathrm{wt}(v)\) of \(v \in C\) counts positions with value 1. A code \(C\) is called doubly even if all \(v \in C\) satisfy \(\mathrm{wt}(v) \equiv 0 \pmod{4}\) [1909.05143, 2601.02332]. If additionally \(C = C^\perp\), the code is self-dual, necessarily of dimension \(n/2\) and admitting only lengths divisible by 8.

The extremal minimum distance for a doubly even self-dual code satisfies the Mallows–Sloane bound:
\[
d \le 4 \left\lfloor \frac{n}{24} \right\rfloor + 4
\]
with equality for extremal codes [1902.08739]. These codes maximize both error-detecting and error-correcting capability among all such codes.

## 2. Algebraic Constructions and Structural Theorems

The algebraic structure of doubly even binary codes is governed by symmetries of the underlying space and the weight properties. Gleason’s theorem and subsequent extensions dictate the permissible weight enumerators. For any doubly even self-dual code of length \(n\),
\[
W_C(y) = \sum_{j=0}^{\left\lfloor n/24 \right\rfloor} b_j\, (1 + 14y^4 + y^8)^{n/8-3j}(y^4(1-y^4)^4)^j
\]
with integer coefficients \(b_j\) [1902.08739].

Doubly evenness requires that the block intersection numbers in associated support designs are always even, ensuring heavy combinatorial structure.

## 3. Constructions: Four-Circulant and Composite-Matrix Methods

Several explicit algebraic constructions exist for doubly even codes, many leveraging automorphism and symmetry.

**Four-circulant construction:** Let \(A, B\) be \(n \times n\) binary circulant matrices. The block generator matrix
\[
G = \begin{pmatrix} I_n & A & B \\ 0 & B^T & A^T \end{pmatrix}
\]
defines a self-dual code iff \(AA^T + BB^T = I_n\), and doubly evenness if all codewords have weight divisible by 4 [1902.08739, 1404.0195].

**Composite matrix/group ring techniques:** Codes can be constructed over rings such as \(\mathbb{F}_2 + u\mathbb{F}_2\) or \(\mathbb{F}_4 + u\mathbb{F}_4\), employing Gray maps to ensure preservation of Lee-weight and duality. For group \(G\), generator matrices constructed via composite block-matrices \(\Omega(v)\) satisfying orthogonality constraints yield new families of extremal doubly even self-dual codes [2106.12355].

See Table 1 for construction types and their parameters:

| Construction Type   | Typical Lengths        | Key Criterion                                        |
|---------------------|-----------------------|------------------------------------------------------|
| Four-circulant      | 72, 80, 96, 112, 120, 128 | \(AA^T + BB^T = I_n\); weights divisible by 4         |
| Composite matrix    | 96                    | Block-matrix satisfies \(\Omega(v)\,\Omega(v)^T = I_n\) |
| Duadic extension    | $2^m$–lengths extended | Achieves self-dual, doubly even via parity extension  |

## 4. Classification and Enumeration of Extremal Codes

Classification of extremal doubly even self-dual codes focuses on inequivalence (typically determined by weight enumerators or automorphism group properties) and explicit enumeration.

Harada’s work [1902.08739] produced numerous new extremal codes of lengths 112, 120, 128. For length 112, three inequivalent codes (H₁₁₂, D₁₁₂, E₁₁₂) were identified, distinguished by their distinct multisets of inner-product counts among weight-20 codewords. In length 120, at least 526 inequivalent codes were constructed, distinguished by the parameter \(a\) in their unique weight enumerator forms.

For length 96, both four-circulant and composite-matrix constructions have yielded more than 200 inequivalent doubly even [96,48,16] codes, substantially enriching the space of possible codes [1404.0195, 2106.12355].

## 5. Code Loops and Algebraic Applications

A fundamental application of doubly even codes is the construction of code loops—Moufang loops with a central subgroup \(\{ \pm 1 \}\), where the binary code's structure induces loop multiplication via a cocycle \(\phi\) [1909.05143, 2601.02332]. For \(V \subseteq \mathbb{F}_2^n\) doubly even,
\[
L(V) = \{ \pm 1 \} \times V,
\]
with product \((\varepsilon, u)\cdot(\delta, v) = (\varepsilon\delta\phi(u, v), u+v)\), and the factor set \(\phi\) determined by codeword weights and pairwise intersections.

Minimal and reduced representations of code loops are classified by the degree and the “type” (sorted tuple of equivalence-class sizes among coordinates). For ranks 3 and 4, the types determine the loop up to isomorphism, and minimal code representations have been explicitly constructed [2601.02332].

## 6. Designs and Weight Distribution

Doubly even codes are tightly linked with combinatorial designs through their support structures. The Assmus–Mattson theorem ensures that, for extremal doubly even self-dual codes, the supports of minimum-weight codewords form t-designs, typically with \(t=5\) [1311.2124]. It is shown that higher strength (e.g., \(t=6,7,8\)) is extremely rare, only potentially realized in certain sporadic lengths and never for \(t=8\).

The weight enumerator and block intersection statistics impose strong integrality conditions on possible support designs, forbidding the existence of many hypothetical high-t designs attached to extremal doubly even codes.

## 7. Extensions, Generalizations, and Open Problems

Recent work has generalized extension theorems to more general rings, leading to new families of doubly even self-dual codes (notably via F₂+uF₂ and F₄+uF₄-lifts) and new weight enumerators [1404.0195]. For cyclic and duadic codes, parity extension produces doubly even self-dual codes of arbitrary length \(n+1\) where \(n\) is odd, with root-like lower bounds on minimum distance [2404.18053].

An open direction remains the full classification of code loop representations in higher rank, as well as determining necessary and sufficient conditions for the existence of extremal doubly even codes of specific length and automorphism structure [1511.02972]. The congruence properties among weight distributions for small lengths (notably 24) suggest deeper modular connections [2303.07587].

## References

- [1902.08739]: Harada, “New doubly even self-dual codes having minimum weight 20”
- [1404.0195]: Kaya & Yıldız, “Extension theorems for self-dual codes over rings and new binary self-dual codes”
- [2106.12355]: Yildiz & Kaya, “New binary self-dual codes of lengths 80, 84 and 96 from composite matrices”
- [2601.02332]: Pires, Grishkov, Rasskazova, “Representations of code loops by binary codes”
- [1311.2124]: Harada, “On the support designs of extremal binary doubly even self-dual codes”
- [2404.18053]: Li, Tang, Ding, “Binary duadic codes and their related codes with a square-root-like lower bound”
- [2303.07587]: Nagaoka & Oura, “Note on the Type II codes of length $24$”
- [1511.02972]: Harada & Munemasa, “On $s$-extremal singly even self-dual $[24k+8,12k+4,4k+2]$ codes”

Source: https://www.emergentmind.com/topics/doubly-even-binary-codes