Sequences of LCD AG codes and LCP of AG Codes attaining the Tsfasman-Vladut-Zink bound

Adler Marques; Luciane Quoos

Sequences of LCD AG codes and LCP of AG Codes attaining the Tsfasman-Vladut-Zink bound

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Pith's one-line read Explicit infinite LCD and LCP code families over $F_{q^2}$, built from the Garcia–Stichtenoth tower, attain the Tsfasman–Vladut–Zink bound and beat the Gilbert–Varshamov bound for large $q$.

desk verdict Explicit LCD/LCP/self-dual AG code families attaining the TVZ bound on the Garcia-Stichtenoth tower; the construction is sound, and the stress-test worry about Proposition 4.2 dissolves once you note ℓ(B)=1. read the letter →

arxiv 2505.23937 v1 pith:WZFUW7OQ submitted 2025-05-29 math.AG cs.ITmath.IT

Adler Marques , Luciane Quoos This is my paper

classification math.AGcs.ITmath.IT MSC 14G5011T7194B2714Q05

keywords LCDcodesLCPofalgebraicgeometryGarcia–StichtenothtowerTsfasman–Vladut–Zinkboundself-dualfunctionfieldsGilbert–Varshamov

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs explicit infinite families of error-correcting codes with a structural extra property: each code is complementary to its dual (LCD), or comes in a complementary pair (LCP), two families used in cryptography against side-channel and fault attacks. Working over the field $F_{q^2}$ with $q \geq 4$, the authors use each level of the Garcia–Stichtenoth tower of function fields to define a code from a suitably chosen divisor, and they describe the needed non-special divisors of small degree at every level. They prove these codes have rate $R$ and relative distance $\delta$ whose limits satisfy $R + \delta \geq 1 - 1/(q-1)$, the Tsfasman–Vladut–Zink bound, so for large $q$ they beat the Gilbert–Varshamov bound constructively rather than by an existence argument. The same tower is also shown to produce asymptotically good self-orthogonal codes for $q \geq 7$ and, when $q$ is even, self-dual codes for $q \geq 8$.

What carries the argument

The machinery is the Garcia–Stichtenoth tower $T_m$ over $F_{q^2}$ together with explicit non-special divisors of degree $g_m - 1$ on every level. The tower supplies many rational places, with limit $\lambda(T) = q-1$, and a fully described ramification divisor; the key identity is the description of the canonical divisor $W(m) = (dt/t)$ in Lemma 5.3, which lets the authors write the dual of $C_L(D(m),G(m))$ as $C_L(D(m), D(m) - G(m) + W(m))$. Choosing $G(m)$ and $H(m)$ with prescribed gcd and sum (or least common multiple) then forces the LCD and LCP conditions, and the explicit degrees of these divisors give the limiting rate and distance.

What would settle it

Compute the first nontrivial case, say $q=4$, $m=2$, inside the tower: write $T_2$ as an Artin–Schreier extension of $F_{16}(x_1)$, form $D(2)$ as the zero divisor of $t = (x_1^{16} - x_1)/(x_1^4 + x_1)$, take $G(2)$ and $H(2)$ from Theorem 5.5, and check with a computer algebra system whether $C_L(D(2),G(2))^\perp$ equals $C_L(D(2),H(2))$ and whether its dimension is $24$; a mismatch would disprove the claimed construction.

Watch

Extended reading notes

Core claim

The authors claim that for every $q \geq 4$ there are explicit infinite sequences of LCD codes over $F_{q^2}$, and explicit infinite sequences of LCP pairs, both obtained from the Garcia–Stichtenoth tower and both attaining the TVZ bound $R + \delta \geq 1 - 1/(q-1)$. The construction attaches to each level $T_m$ the divisor $D(m)$ equal to the zero divisor of $t = (x_1^{q^2} - x_1)/(x_1^q + x_1)$, then chooses divisors $G(m)$ and $H(m)$ whose gcd is a non-special divisor of degree $g_m - 1$ and whose sum (or least common multiple) is controlled by the canonical divisor $W(m) = (dt/t)$. Under those divisor conditions, Theorem 5.4 makes $C_L(D(m),G(m))$ an LCD code with dual $C_L(D(m),H(m))$, while Theorem 5.6 makes the pair an LCP; Theorem 5.5 and Theorem 5.7 compute the resulting lengths, dimensions, and distance lower bounds, and pass to the limit to reach the TVZ bound. The paper further derives self-orthogonal and, for even $q$, self-dual sequences from the same tower.

Load-bearing premise

The construction rests on the claimed formula for how the places of the tower split and ramify at each level (Lemma 5.2); if that ramification formula were wrong, the canonical divisor used to form duals would be wrong and the LCD and LCP divisor conditions would no longer hold.

Editorial extensions

If this is right

For every $q \geq 4$ there is an infinite sequence of LCD codes over $F_{q^2}$ whose asymptotic parameters satisfy $R + \delta \geq 1 - 1/(q-1)$; for large $q$ this is beyond the Gilbert–Varshamov curve.
The same tower gives infinite sequences of LCP pairs $(C_L(D(m),G(m)), C_L(D(m),H(m)))$ in which both component codes individually attain the TVZ bound.
The tower also yields self-orthogonal code sequences meeting the TVZ bound for $q \geq 7$, and for even $q \geq 8$ self-dual sequences with rate $1/2$ that exceed the Gilbert–Varshamov bound.
All constructions are explicit: each code is given by divisors written down on the $m$-th level of the tower, so the families are effective rather than existential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The divisor recipe is portable: any tower with the same quality of ramification data, namely many rational places and an explicit different divisor, should yield analogous LCD and LCP families; the missing ingredient is usually writing explicit non-special divisors of degree $g-1$ at every level.
The proof gives explicit limiting rates and distances, so one could compute the smallest $q$ at which these families exceed the Gilbert–Varshamov bound; the paper states that this happens for sufficiently large $q$ but does not identify the threshold.
Because the codes are given by explicit divisors on an optimal tower, the construction could be turned into concrete code tables for modest $q$ and $m$, for instance as test cases for cryptographic countermeasures, though the paper does not address implementation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper earns its place: it turns existential results for LCD and LCP codes meeting the TVZ bound into explicit families on the Garcia-Stichtenoth tower. The key technical work is the construction of non-special divisors of degree g_m−1 (Prop. 4.2) and the explicit canonical divisor computation (Lemma 5.3). I spot-checked the gcd/lmd identities in Theorems 5.5 and 5.7, the degree calculations, and the limiting rates; they are internally consistent. The asymptotic claims R+δ = 1−1/(q−1) for both families check out.

The stress-test worry about subtracting P∞ from B does not hold up. The proof shows B is linearly equivalent to a non-special divisor of degree g_m with ℓ=1, so B itself has ℓ(B)=1; hence |B| is exactly {B}. Since P∞ is not in Supp(B), B−P∞ is automatically non-special: the only possible effective divisor in |B−P∞| would be B−P∞, which is not effective. The paper could have spelled this out, but the step is valid.

Soft spots are minor. First, the paper never explicitly verifies Supp(G)∩Supp(D)=∅ in Theorems 5.5 and 5.7. It is true: D is the zero divisor of t, whose support consists of completely split places, while G and H are supported on the ramified places (the Pα, P∞, and the A_k). Still, the authors should state it. Second, Lemma 5.2 imports the different-divisor formula from [15] and [1] without proof. This is acceptable—the cited computations are published—but the lemma is load-bearing, so a referee should confirm the citations are correct. Third, the presentation has a few typos and minor index issues, but nothing that obscures the argument.

Who gets value: coding theorists interested in explicit asymptotically good structured codes, and people working on side-channel-resistance applications of LCD/LCP codes. It also extends the authors' earlier work on Kummer/hyperelliptic/elliptic fields to the optimal tower. I'd send this to a serious referee; it deserves a careful look, and I expect it to be accepted after minor revisions.

Referee Report

0 major / 6 minor

Summary. The paper constructs explicit infinite sequences of LCD codes and of linear complementary pairs (LCPs) over F_{q^2} obtained from the Garcia-Stichtenoth tower. For q>=4 (and q>2 for LCPs), the main theorems (Theorems 5.5 and 5.7) give divisors G(m), H(m) such that the associated AG codes have limit parameters satisfying R + delta >= 1 - 1/(q-1), hence attain the Tsfasman-Vladut-Zink bound. The key technical ingredients are explicit non-special divisors of degree g_m - 1 on each tower level (Proposition 4.2), a computation of the canonical divisor W(m) = (dt/t) (Lemma 5.3), and the application of LCP/LCD criteria from prior work (Theorem 2.8). Section 6 additionally constructs self-orthogonal sequences (q >= 7) and self-dual sequences (q even, q >= 8) from the same tower, also attaining the TVZ bound.

Significance. If correct, this is a valuable contribution: it provides fully explicit infinite families of LCD codes and LCPs that attain the TVZ bound, going beyond existential results by Carlet et al. and Jin-Xing. The construction is concrete, with closed-form divisors and transparent asymptotic computations. The paper also gives explicit self-orthogonal and self-dual sequences with the same asymptotic optimality. The main proofs are computational and checkable, and the reliance on published ramification and Weierstrass-semigroup results is clearly indicated, making the paper a useful reference for explicit good AG code families.

minor comments (6)

[Section 4, Proposition 4.2] The step 'from [3, Lemma 3]' is not self-contained. Subtracting an arbitrary rational place from a non-special divisor of degree g does not in general preserve non-speciality, and the paper only notes that P_infty is not in the support of B. The conclusion is nevertheless correct here because B = sum (q^{ceil(m/2)} - 1) A_k is effective and non-special of degree g_m, so ell(B)=1 and L(B)=F_q; hence ell(B - P_infty)=0, which is equivalent to B - P_infty being non-special. Please include this argument (or state the precise lemma from [3] and verify its hypotheses) to make the proof complete.
[Section 5, Theorem 5.6] The proof verifies that gcd(G,H) and lmd(G,H) - D are non-special of degree g_m - 1, but it does not explicitly verify condition (ii) of Theorem 2.8, namely ell(G) + ell(H) = n_m. This condition follows from the non-speciality of G and H (via G,H >= gcd), the relation G+H = gcd + lmd, and deg(lmd) = n_m + g_m - 1, but the argument should be stated.
[Section 5, Theorems 5.5 and 5.7] The condition Supp(G(m)) cap Supp(D(m)) = empty (and similarly for H(m)) is not explicitly verified. It is true by construction because D(m) is supported on places lying over F_{q^2} \setminus Omega, while G(m) and H(m) are supported on places over Omega, on zeros of x_1, and on P_infty; please state this for completeness.
[Abstract and Introduction] The claim that for sufficiently large q the constructed codes exceed the Gilbert-Varshamov bound is stated in the abstract but not proved or referenced in the main theorems. A brief sentence indicating the standard threshold (e.g., where the TVZ line lies above the GV curve for F_{q^2}) would help the reader.
[Proof of Theorem 5.7] In the display for delta_m, there is an unmatched closing parenthesis in the numerator (after 'q^{ceil(m/2)}'), and the limit expression '1 - 3q - 2/(q^2 - q)' should be typeset with parentheses around the numerator as (3q-2)/(q^2-q).
[Lemma 5.3] The reference to Stichtenoth's remark appears as '[26, Remark 4.3.7, (c)]' in one place and '[26, Remark 4.4.7 (c)]' in another; please unify the citation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the LCD/LCP constructions are derived from independent external results and explicit divisor computations, with only minor non-load-bearing self-citation.

full rationale

The central claim — explicit infinite sequences of LCD codes and LCPs over F_{q^2} attaining the TVZ bound — is not equivalent to any fitted parameter or to the paper's own inputs. The divisors G(m) and H(m) are explicit, their degrees are computed directly, and the limiting rates and distances are evaluated arithmetically against the independent Garcia-Stichtenoth tower having lambda(T) = q - 1. The external benchmarks (TVZ line, GV bound) are not used to choose any constant. The paper cites its own prior work [10] and [11] only as motivation ('Based on the ideas in [10]'), while the technical non-special divisor construction and canonical divisor computation are carried out in this paper from the known tower data in [15], [21], and [1]. The most delicate step, Proposition 4.2, invokes the external lemma [3, Lemma 3] to subtract P_infinity from a non-special divisor; whether all hypotheses of that lemma are verified is a proof-gap concern, not a circularity, since the lemma is not the paper's own conclusion restated. Similarly, Lemma 5.2 assembles ramification exponents from published sources [15] and [1]; if incorrect, the canonical divisor would fail, but that is an imported factual input rather than a self-referential derivation. Overall, the derivation chain is self-contained relative to standard external results, and the self-citations are not load-bearing.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants or invented entities. It rests on standard algebraic geometry and on cited properties of the Garcia-Stichtenoth tower. The novel part is the explicit non-special divisor construction in Section 4, which is proved in the paper. The heaviest external load is the ramification and different formula in Lemma 5.2, drawn from [1] and [15].

assumptions (6)

standard math Riemann-Roch theorem and the dimension formula ℓ(G) = deg(G) + 1 - g for non-special divisors.
Invoked in Proposition 2.5 and Theorem 5.4 to compute code dimensions.
standard math Garcia-Stichtenoth tower facts: genus formula g_m, split and ramified places, and λ(T) = q - 1.
Section 3; these give n_m = q^m(q - 1) rational places and the ramification pattern.
standard math The divisor (q^m - q^{⌈m/2⌉})P_∞ - A(m) is non-special of degree g_m [21].
Used in Proposition 4.2 to build non-special divisors of degree g_m - 1.
domain assumption The different exponents of the tower satisfy d(P_α) = d(P_∞) = 2(q^{m-1} - 1) and d(Q) = 2(q^{m-2k-1} - 1) for Q in A_k.
Lemma 5.2 rests on this; it determines the canonical divisor W(m) and hence the dual code.
standard math The LCP criterion of Bhowmick-Dalai-Mesnager [6, Theorem 3.5] and the dual formula CL(D,G)⊥ = CL(D, D-G+W).
Theorems 5.4 and 5.6 use these to certify LCD and LCP.
standard math Subtracting a rational point not in the support of a non-special divisor of degree g preserves non-speciality [3, Lemma 3].
Proposition 4.2 relies on this to decrease degree from g_m to g_m - 1.

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Pith. "Pith review of Sequences of LCD AG codes and LCP of AG Codes attaining the Tsfasman-Vladut-Zink bound." pith.science (2026). https://pith.science/paper/WZFUW7OQ

@misc{pith2026250523937,
  author       = {Pith},
  title        = {Pith review of: Sequences of LCD AG codes and LCP of AG Codes attaining the Tsfasman-Vladut-Zink bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZFUW7OQ}},
  note         = {Machine review of arXiv:2505.23937}
}

abstract

Since Massey introduced linear complementary dual (LCD) codes in 1992 and Bhasin et al. later formalized linear complementary pairs (LCPs) of codes - structures with important cryptographic applications - these code families have attracted significant interest. We construct infinite sequences $(C_i)_{i \geq 1}$ of LCD codes and of LCPs $(C', D')_{i \geq 1}$ over $\mathbb{F}_{q^2}$ obtained from the Garcia-Stichtenoth tower of function fields, where we describe suitable non-special divisors of small degree on each level of the tower. These families attain the Tsfasman-Vl\u{a}du\c{t}-Zink bound and, for sufficiently large $q$ exceed the classic Gilbert-Varshamov bound, providing explicit asymptotically good constructions beyond existential results. We also exhibit infinite sequences of self-orthogonal over $\mathbb{F}_{q^2}$ and, when $q$ is even, self-dual codes from the same tower all meeting the Tsfasman-Vl\u{a}du\c{t}-Zink bound.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation
math.AG 2026-07 conditional novelty 7.0 of 10

A degree test on Mumford polynomials characterizes non-special divisors on hyperelliptic curves and yields explicit LCP/LCD code constructions, with a verified q=5 MDS LCD example.

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Works this paper leans on

27 extracted references · 27 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [15]

Garcia and H

A. Garcia and H. Stichtenoth , On the Asymptotic Behaviour of Some Towers of Function Field s over Finite Fields, Journal of Number Theory, 61 (1996), pp. 248–273

work page 1996

[2] [1]

Aleshnikov, V

I. Aleshnikov, V. Deolalikar, P. V. Kumar, and H. Stichtenoth , Towards a Basis for the Space of Regular Functions in a Tower of Function Fields Meeting the Drinfeld -Vladut Bound , in Finite Fields and Applications, D. Jungnickel and H. Niederreiter, eds., Berlin, Heidelber g, 2001, Springer Berlin Heidelberg, pp. 14–24

work page 2001

[3] [2]

Ballet and M

S. Ballet and M. Koutchoukali , On the non-special divisors in algebraic function ﬁelds deﬁ ned over ﬁnite ﬁelds , Polynesian Journal of Mathematics, 2 (2025), pp. 1–41

work page 2025

[4] [3]

Ballet and D

S. Ballet and D. Le Brigand , On the existence of non-special divisors of degree g and g − 1 in algebraic function ﬁelds over Fq, Journal of Number Theory, 116 (2006), pp. 293–310

work page 2006

[5] [4]

Bassa and H

A. Bassa and H. Stichtenoth , Self-dual codes better than the Gilbert–Varshamov bound , Designs, Codes and Cryptography, 87 (2019), pp. 173–182

work page 2019

[6] [5]

Beelen , A survey on recursive towers and Ihara’s constant , in Curves over ﬁnite ﬁelds—past, present and future, vol

P. Beelen , A survey on recursive towers and Ihara’s constant , in Curves over ﬁnite ﬁelds—past, present and future, vol. 60 of Panor. Synthèses, Soc. Math. France, Pari s, 2023, pp. 67–94

work page 2023

[7] [6]

Bhowmick, D

S. Bhowmick, D. K. Dalai, and S. Mesnager , On linear complementary pairs of algebraic geometry codes o ver ﬁnite ﬁelds , Discrete Mathematics, 347 (2024), p. 114193

work page 2024

[8] [7]

Bringer, C

J. Bringer, C. Carlet, H. Chabanne, S. Guilley, and H. Maghrebi , Orthogonal Direct Sum Masking , in Information Security Theory and Practice. Securing the Int ernet of Things, D. Naccache and D. Sauveron, eds., Berlin, Heidelberg, 2014, Springer Berlin Heidelberg, pp. 40–56

work page 2014