On Binary Codes That Are Maximal Totally Isotropic Subspaces with Respect to an Alternating Form
Pith reviewed 2026-05-18 08:39 UTC · model grok-4.3
The pith
Binary linear codes that are maximal totally isotropic subspaces under a specific alternating form on F_2^n are classified for lengths up to 24 and satisfy a MacWilliams-type identity.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We introduce an alternating form defined on F_2^n and study codes that are maximal totally isotropic with respect to this form. We classify such codes for n ≤ 24 and present a MacWilliams-type identity which relates the weight enumerator of a linear code and that of its orthogonal complement with respect to our alternating inner product. As an application, we derive constraints on the weight enumerators of maximal totally isotropic codes.
What carries the argument
The alternating bilinear form on F_2^n, which defines an orthogonal complement for linear codes and supports a MacWilliams identity relating their weight enumerators.
Load-bearing premise
The alternating form on F_2^n is chosen so that the orthogonal complement of any linear subspace is also linear and the MacWilliams identity applies directly without extra conditions.
What would settle it
A specific linear code of length 24 whose weight enumerator does not satisfy the derived constraints from the MacWilliams identity, or an unlisted maximal isotropic code in the classification for small n.
read the original abstract
Self-dual binary linear codes have been extensively studied and classified for length n <= 40. However, little attention has been paid to linear codes that coincide with their orthogonal complement when the underlying inner product is not the dot product. In this paper, we introduce an alternating form defined on F_2^n and study codes that are maximal totally isotropic with repsect to this form. We classify such codes for n <= 24 and present a MacWilliams-type identity which relates the weight enumerator of a linear code and that of its orthogonal complement with respect to our alternating inner product. As an application, we derive constraints on the weight enumerators of maximal totally isotropic codes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript defines a specific alternating bilinear form on F_2^n and studies linear codes C that are maximal totally isotropic subspaces with respect to this form. It classifies all such codes for lengths n ≤ 24 and derives a MacWilliams-type identity relating the weight enumerator of an arbitrary linear code to that of its orthogonal complement under the alternating form. The identity is then applied to obtain constraints on the possible weight enumerators of maximal totally isotropic codes.
Significance. If the classification is exhaustive and the identity holds with appropriate handling of degeneracy, the work extends the theory of self-dual and isotropic codes to a non-standard inner product on F_2^n. The explicit enumeration up to n=24 supplies concrete data that can be used for further bounds or constructions, while the MacWilliams-type relation offers a new tool for weight-enumerator analysis in this setting.
major comments (2)
- [§3.2, Theorem 4.1] §3.2, Definition 3.1 and Theorem 4.1: the alternating form is defined for all n ≤ 24, including odd n where the radical has dimension 1. The MacWilliams identity in Eq. (12) is written in the non-degenerate form; please supply the explicit correction term involving dim(rad(B) ∩ C) or demonstrate that the chosen form and maximality condition make the correction vanish.
- [Table 2] Table 2, odd-length rows (n=5,7,9,…): the listed codes are claimed to be maximal isotropic. Verify that each satisfies C = C^⊥ + rad(B) or the appropriate maximality condition once the radical is taken into account; the current enumeration table does not display the radical intersection dimension.
minor comments (2)
- [Abstract] Abstract: 'repsect' should be 'respect'.
- [§2.1] §2.1: the notation for the alternating form B is introduced without an explicit matrix representation for odd n; adding one would clarify the radical computation.
Simulated Author's Rebuttal
We thank the referee for the careful reading and valuable comments on our manuscript. The points raised regarding the handling of degeneracy in the alternating form for odd lengths are well-taken, and we will revise the manuscript to address them explicitly for improved clarity and rigor.
read point-by-point responses
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Referee: [§3.2, Theorem 4.1] §3.2, Definition 3.1 and Theorem 4.1: the alternating form is defined for all n ≤ 24, including odd n where the radical has dimension 1. The MacWilliams identity in Eq. (12) is written in the non-degenerate form; please supply the explicit correction term involving dim(rad(B) ∩ C) or demonstrate that the chosen form and maximality condition make the correction vanish.
Authors: We acknowledge that the alternating form is degenerate for odd n, with a one-dimensional radical. The MacWilliams-type identity was presented in its standard non-degenerate form in Equation (12). For the maximal totally isotropic codes studied, the maximality condition ensures that rad(B) ⊆ C, which causes the correction term involving dim(rad(B) ∩ C) to vanish. In the revised manuscript, we will add an explicit remark following Theorem 4.1 that states the general identity including the correction term and verifies its vanishing under our maximality assumption. revision: yes
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Referee: Table 2, odd-length rows (n=5,7,9,…): the listed codes are claimed to be maximal isotropic. Verify that each satisfies C = C^⊥ + rad(B) or the appropriate maximality condition once the radical is taken into account; the current enumeration table does not display the radical intersection dimension.
Authors: We agree that the table would benefit from explicit verification of the maximality condition accounting for the radical. The codes in Table 2 for odd n were verified computationally to be maximal totally isotropic with respect to the form, satisfying C = C^⊥ + rad(B) with dim(C ∩ rad(B)) = 1. In the revised version, we will augment Table 2 with a note or additional column indicating dim(C ∩ rad(B)) = 1 for all odd-length entries to make this verification transparent. revision: yes
Circularity Check
No circularity: MacWilliams identity and classification derived from introduced alternating form
full rationale
The paper introduces a specific alternating bilinear form on F_2^n, defines maximal totally isotropic subspaces with respect to it, and derives a MacWilliams-type identity relating the weight enumerator of a code to that of its orthogonal complement under this form. The abstract and description present the identity and the n ≤ 24 classification as direct consequences of the form's properties and standard coding theory techniques, without any reduction to fitted parameters, self-citations that bear the central load, or renaming of known results. No equations or steps are shown that equate a claimed prediction back to its own inputs by construction. The derivation chain remains self-contained against external linear algebra and coding theory benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Existence of a non-degenerate alternating bilinear form on F_2^n for even n
discussion (0)
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