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arxiv: 2607.02170 · v1 · pith:OF3ARCF2new · submitted 2026-07-02 · 💻 cs.IT · math.IT· quant-ph

Generalized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes

Yang Li , Martianus Frederic Ezerman , Shitao Li , San Ling , Zhonghua Sun This is my paper

Pith reviewed 2026-07-03 05:03 UTC · model grok-4.3

classification 💻 cs.IT math.ITquant-ph
keywords generalized extended codesHermitian hullentanglement-assisted quantum codesEAQECCqubit codesqutrit codesmonomial equivalenceHermitian dual distance
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The pith

Generalized extended codes C(u,a) let researchers independently tune Hermitian hull dimension and dual distance to build better entanglement-assisted quantum codes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that generalized extended codes are monomially equivalent to Hermitian duals of related extended codes from the Hermitian dual of C. It shows every suitable length-(n+1) code is equivalent to some C(u,a) built from a length-n code C. The Hermitian hull dimension of C(u,a) depends on where u sits relative to the sum C + C perp H, while the dual distance depends on how u interacts with the minimum-weight words of C perp H. Explicit conditions are derived that let both quantities be increased at once. These controls produce 267 new entanglement-assisted qubit codes of length at most 40 and 14 new qutrit codes of length at most 25, with confirmed improvements on 236 qubit and 8 qutrit parameter sets.

Core claim

Any generalized extended code is monomially equivalent to the Hermitian dual of a code closely related to a second kind of extended code of C perp H. Every [n+1,k+1] linear code D over F_{q^2} with d(D perp H)>1 is monomially equivalent to C(u,a) for fixed a in F_{q^2}^* and suitable u. The Hermitian hull dimension and Hermitian dual distance of C(u,a) are characterized by the position of u relative to C + C perp H and its interaction with minimum-weight codewords of C perp H. Several conditions allow simultaneous increase of both the hull dimension and the dual distance.

What carries the argument

The generalized extended code C(u,a) of an [n,k] linear code C, formed by adjoining a scalar a and vector u; the construction supplies explicit levers on Hermitian hull dimension and Hermitian dual distance.

If this is right

  • The Hermitian construction for EAQECCs can now be applied to families of C(u,a) whose hull and dual-distance parameters are chosen independently.
  • 267 new entanglement-assisted qubit codes appear for lengths up to 40.
  • 14 new entanglement-assisted qutrit codes appear for lengths up to 25.
  • Improvements over existing tables are confirmed for 236 qubit codes and 8 qutrit codes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same position-based control on hull dimension may extend to other dual-distance measures or to non-Hermitian settings.
  • Tables of best-known EA codes can be updated systematically by enumerating admissible u vectors for each base code C.
  • The ability to raise hull dimension while preserving or increasing dual distance may reduce the entanglement consumption needed for a target error-correcting capability.

Load-bearing premise

The stated monomial equivalences and the characterizations of hull dimension and dual distance for C(u,a) are correct, so the listed new EA codes can actually be built and verified as improvements.

What would settle it

A concrete linear code D of length n+1 whose Hermitian dual distance exceeds 1 but cannot be expressed as any C(u,a) for an [n,k] code C, fixed a, and vector u, or a case where the predicted hull dimension from u's position does not match the actual hull.

Figures

Figures reproduced from arXiv: 2607.02170 by Martianus Frederic Ezerman, San Ling, Shitao Li, Yang Li, Zhonghua Sun.

Figure 1
Figure 1. Figure 1: Reduction of the search space for extension vectors within [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of $\C^{\perp_{\rm H}}$. Every $[n+1,k+1]_{q^2}$ linear code $\D$ with $d(\D^{\perp_{\rm H}})>1$ is monomially equivalent to the generalized extended code $\C({\bf u},a)$ of an $[n,k]_{q^2}$ linear code $\C$ for a fixed $a\in\F_{q^2}^{*}$ and some ${\bf u}\in\F_{q^2}^{n}$. We then characterize the Hermitian hull and Hermitian dual distance of $\C({\bf u},a)$ in terms of the position of ${\bf u}$ relative to $\C+\C^{\perp_{\rm H}}$ and the interaction between ${\bf u}$ and the minimum weight codewords of $\C^{\perp_{\rm H}}$, respectively. We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of $\C({\bf u},a)$. In particular, several conditions for simultaneously increasing the Hermitian hull dimension and the Hermitian dual distance of $\C({\bf u},a)$ are derived. Applying these results to the Hermitian construction for EAQECCs gives us $267$ new EA qubit codes of lengths $n \leq 40$ and $14$ new EA qutrit codes of lengths $n \leq 25$ compared to the best-known codes in Grassl's code tables and the imporvements recorded in very recent works in the literature. Among the new parameter sets, we confirm improvements for $236$ qubit and $8$ qutrit codes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves that generalized extended codes are monomially equivalent to Hermitian duals of related extended codes, shows that every suitable [n+1,k+1]_{q^2} code is monomially equivalent to some C(u,a), characterizes the Hermitian hull dimension of C(u,a) via the position of u relative to C + C^⊥_H and the Hermitian dual distance via interaction of u with minimum-weight words of C^⊥_H, derives explicit criteria to control these quantities (including conditions for simultaneously increasing both), and applies the results via the Hermitian EA construction to report 267 new EA qubit codes (n≤40) and 14 new EA qutrit codes (n≤25), confirming improvements for 236 qubit and 8 qutrit parameter sets over Grassl tables and recent literature.

Significance. If the characterizations hold, the work supplies a constructive method for tuning hull dimension and dual distance in the Hermitian EA framework, directly yielding new code parameters that enlarge the known tables; this is a concrete contribution to the catalog of entanglement-assisted quantum codes.

major comments (2)
  1. [Abstract (and the section stating the characterizations)] The central claim of 267/14 new EA codes (and the 236/8 confirmed improvements) rests entirely on the correctness of the hull-dimension characterization (position of u relative to C + C^⊥_H) and dual-distance characterization (interaction of u with min-weight words of C^⊥_H) together with the derived simultaneous-increase criteria; any off-by-one or missed case in these statements would mean the constructed C(u,a) fails to achieve the claimed parameters, rendering the listed improvements invalid.
  2. [The equivalence theorems] The monomial-equivalence statements (any generalized extended code ≡ Hermitian dual of a related extended code; every suitable D ≡ C(u,a) for fixed a and some u) are load-bearing for the reduction to the Hermitian EA construction; the manuscript must make explicit whether these equivalences preserve the Hermitian inner product and the relevant distances.
minor comments (2)
  1. [Abstract] Abstract contains the typo "imporvements" (should be "improvements").
  2. [Introduction / definitions] Notation for the generalized extended code C(u,a) and the fixed scalar a should be introduced with a short definition or reference to the precise construction before the equivalence and characterization statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of its significance. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract (and the section stating the characterizations)] The central claim of 267/14 new EA codes (and the 236/8 confirmed improvements) rests entirely on the correctness of the hull-dimension characterization (position of u relative to C + C^⊥_H) and dual-distance characterization (interaction of u with min-weight words of C^⊥_H) together with the derived simultaneous-increase criteria; any off-by-one or missed case in these statements would mean the constructed C(u,a) fails to achieve the claimed parameters, rendering the listed improvements invalid.

    Authors: The hull-dimension and dual-distance characterizations, together with the simultaneous-increase criteria, are established by direct case analysis on the position of u relative to C + C^⊥_H and on the Hermitian inner products of u with the minimum-weight vectors of C^⊥_H; the proofs appear in Theorems 3.1–3.3 and Corollary 3.4. These arguments enumerate all possible configurations of the relevant inner products and therefore contain no omitted cases. The 267/14 new EA codes are obtained by applying the criteria to explicit base codes whose parameters are independently verified. To increase transparency we will insert, in the revised manuscript, a short additional paragraph that restates the exhaustive case division and supplies two further low-length numerical checks. revision: partial

  2. Referee: [The equivalence theorems] The monomial-equivalence statements (any generalized extended code ≡ Hermitian dual of a related extended code; every suitable D ≡ C(u,a) for fixed a and some u) are load-bearing for the reduction to the Hermitian EA construction; the manuscript must make explicit whether these equivalences preserve the Hermitian inner product and the relevant distances.

    Authors: The monomial maps in Theorems 2.1 and 2.2 are constructed explicitly: a coordinate permutation together with component-wise multiplication by nonzero field elements whose conjugates are chosen so that the Hermitian inner product is preserved (i.e., the image of the dual is the dual of the image). Because monomial equivalence preserves Hamming weights, the minimum distances are likewise preserved. We will add, immediately after the statements of Theorems 2.1 and 2.2, a single sentence that records these preservation properties and refers the reader to the explicit form of the monomials given in the proofs. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic derivations are independent of outputs

full rationale

The paper derives monomial equivalence statements and explicit criteria for Hermitian hull dimension and dual distance of C(u,a) directly from linear-algebraic properties over finite fields (position of u relative to C + C^⊥_H and interaction with minimum-weight words). These are then applied to the Hermitian EA construction to enumerate new parameter sets, which are checked for improvement against external tables (Grassl and recent literature). No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the characterizations are presented as proved statements whose validity is required for the new codes but are not presupposed by the construction itself. This is a standard non-circular proof-and-application structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of linear codes, Hermitian duality, and monomial equivalence over finite fields; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard properties of linear codes over finite fields F_{q^2} and their Hermitian duals
    Invoked throughout the characterizations of hull and dual distance.
  • standard math Monomial equivalence preserves the relevant code parameters
    Used to relate generalized extended codes to Hermitian duals.

pith-pipeline@v0.9.1-grok · 5870 in / 1421 out tokens · 32856 ms · 2026-07-03T05:03:34.447353+00:00 · methodology

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Reference graph

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