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Locally Recoverable Codes with availability from a family of fibered surfaces

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Pith's one-line read Fibered surfaces give locally recoverable codes with availability 2, dimension r(r−1)−1, and a sharp distance bound at r=3.

desk verdict New LRC family with availability 2, but the existence of the evaluation set rests on a Chebotarev step that does not close as written. read the letter →

arxiv 2512.08100 v2 pith:NDWOVIF2 submitted 2025-12-08 math.AG cs.ITmath.IT

Cecília Salgado , Lara Vicino This is my paper
classification math.AGcs.ITmath.IT MSC 14H0514G5094B27
keywords LocallyRecoverableCodesavailabilityAGone-variablefunctionfieldsfiberedsurfacesK3Newtonarcsminimumdistance
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 asks whether algebraic surfaces that fiber over a line in two different ways can be used to build locally recoverable codes — codes in which every symbol can be recovered from a small number (r) of other symbols, and indeed from two independent such sets. The answer is yes for a specific family of surfaces E_r and curves M_r embedded in them: evaluating a carefully chosen space of polynomials on a set of 'nice' rational points gives codes with length b(r+1)^2, dimension r(r−1)−1, locality r, and availability 2. The authors prove a lower bound on the minimum distance, and show that for locality r=3 the bound is sharp; for the single-block case b=1 the distance is exactly 8. A reader interested in distributed storage would care because availability 2 means a failed node can still be repaired even if one recovery route is blocked. The construction is also the first to use a doubly elliptic K3 surface as the geometric home of an error-correcting code.

What carries the argument

The machinery is the pair of fibrations on E_r: projection to t and projection to x, together with the curve M_r, a multisection of degree r+1 of both. The two fibrations supply the two recovery sets: fixing x gives a horizontal fiber, fixing t gives a vertical fiber. The function space L has degree less than r in each variable, so on each fiber the values of a function are governed by a Vandermonde system, which is what makes recovery by interpolation possible. The minimum-distance estimate is extracted from the Newton arc of the polynomial P_t(T) defining M_r at the place t=∞, which determines the valuations of elements of L at the poles; the place P_1 is a common zero of all functions in

What would settle it

Check whether the set G_m of nice elements is non-empty for a triple (q, r, m) that just satisfies q^m/m ≥ 2(r+1)!, e.g., r=3, q=5, m=4, by direct search in F_{5^4} for a t such that P_t(T) = T^4 + 2T^2 − T^3 + T^2(t^4+1) − T t^4 + 1 splits into four distinct linear factors. If none exists, the construction has no evaluation points.

Watch

Extended reading notes

Core claim

Take an odd integer r ≥ 3, a prime power q ≡ 1 (mod r+1), and a finite field F_{q^m}. On the surface E_r: y^2 = x^3 − x^2(t^{r+1}+1) + x t^{r+1}, single out the curve M_r given by y = x^{(r+1)/2}+1. Call a rational point P on M_r 'nice' if its horizontal fiber (fixed x) and vertical fiber (fixed t) each contain r further F_{q^m}-rational points of M_r. Choose b orbits of such nice points and evaluate the space L of polynomials x^i t^j with i=1,…,r−2, j=0,…,r−1 together with x^{r−1}t^h, h=0,…,r−2, on the resulting set S_B of b(r+1)^2 points. The paper proves the image is a code with dimension r(r−1)−1, locality r, availability 2, and minimum distance at least n − (2r^2 − 2r − 3); for r=3 this

Load-bearing premise

The load-bearing premise is that the bound in Lemma 3.4 really guarantees at least one completely splitting place, but its proof drops a Chebotarev error term O(q^{m/2}/m) whose size relative to the main term is not controlled by the stated inequality.

Editorial extensions

If this is right

  • For b=1, every code C(B,L) has minimum distance exactly 8, independent of r.
  • For r=3 and b≥2, the lower bound n − (2r^2 − 2r − 3) is attained in the explicit examples computed in Section 4.3.
  • The construction provides the first example of an error-correcting code built from a doubly elliptic K3 surface.
  • Each symbol of a codeword has two disjoint recovery sets of size r, so losing one recovery set does not compromise locality.

Reading between the lines

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

  • A natural next step is to search, for small q and r, for 'nice' elements t of F_{q^m} when q^m/m is only slightly above 2(r+1)!; the Chebotarev bound used in the paper likely overestimates the smallest m, and a computational census could show the true threshold and whether the construction works well beyond the stated guarantee.
  • Because the two recovery mechanisms are independent (Vandermonde interpolation on fibers versus elliptic-curve group law on the K3 model), one could vary the space L or the curve M_r while keeping the two fibrations, obtaining availability-2 LRC families with different rate–distance tradeoffs.
  • The role of the common zero P_1 of all functions in L in the distance bound suggests a design principle: for a fibered-surface construction, choosing L with a large common zero outside the evaluation set may improve the lower bound; this is implicit in the paper's −2 correction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs algebraic-geometry codes from a family of fibered surfaces. For odd r ≥ 3 and q ≡ 1 mod r+1, the authors define a surface E_r and a curve M_r, choose b(r+1)^2 evaluation points obtained from completely splitting places, and evaluate the bidegree-restricted space L defined in Eq. (18). They claim the resulting codes C(B,L) have length b(r+1)^2, dimension r(r−1)−1, locality r, availability 2, and minimum distance d ≥ n − (2r^2 − 2r − 3). For r = 3 the bound is claimed to be sharp, and for b = 1 the distance is claimed to be exactly 8. Section 5 reinterprets the r = 3 case in terms of two genus-1 fibrations on a K3 surface.

Significance. If the central construction and the distance estimates are correct, the paper gives a new infinite family of LRCs with availability 2 from a geometric source different from previous constructions. The recovery procedure via Vandermonde systems is explicit and clean, the lower distance bound is derived directly from pole divisors rather than fitted to examples, and the r = 3 examples in Table 1 provide independent brute-force support. The interpretation via a doubly elliptic K3 surface is attractive and, if properly substantiated, would be a novel contribution. However, two load-bearing points currently lack rigorous proofs: the existence of evaluation points and the exact d = 8 statement.

major comments (3)
  1. [3, Lemma 3.4] The existence of a nice t-bar is not proved. The proof writes the number of completely splitting places as q^m/(m|G|) + O(q^{m/2}/m) and then drops the error term. From q^m/m ≥ 2(r+1)! one only obtains q^m/(m|G|) ≥ 2; the O-term can be much larger, e.g. r = 7, q = 9, m = 6 gives q^m/(m(r+1)!) ≈ 2.2 and q^{m/2}/m ≈ 121.5. Moreover, the argument excludes only the pole of t; if the completely splitting place is t^{r+1} = 0, then t-bar = 0 and the set H_P in Eq. (9) has size 1, so the claimed (r+1)^2 evaluation grid cannot be formed. Since Lemma 3.4 is the only nonemptiness statement for S_B, the code C(B,L) may be undefined.
  2. [4.2, Proposition 4.10] The exact-distance proof for b = 1 is incomplete. After displaying f_min in Eq. (34), the proof asserts that no function in L has more than r^2 + 2r − 7 zeros in S_t-bar, but the supporting 'maximality' argument is informal: it assumes the extremal configuration must consist of r − 1 full vertical fibers together with r − 3 additional horizontal fibers. Partial vanishing patterns are not ruled out. The claim may be true, and Table 1 supports it for r = 3, but the general odd-r case requires a rigorous extremal argument. Since d = 8 is one of the headline parameter claims, this gap is load-bearing.
  3. [5, Lemma 5.2] The alternative recovery proof rests on several assertions that are not fully justified: the statement that the trace of the image of M in the jacobian fibration is contained in the 2-torsion of the generic fiber; the application of [24, Thm 2.4]; and the conclusion from [26, Specialization thm.] that the sum of the four intersection points lies in the 2-torsion of the fiber. Because this section underpins the abstract's K3-surface claim, these statements need either precise proofs with all definitions or an explicit disclaimer that Section 5 is only a sketch. The code parameters themselves are already established by Lemma 4.2.
minor comments (4)
  1. [Throughout] Several cross-references are to 'Theorem 2.3', 'Theorem 3.4', 'Theorem 3.7', etc., where the cited items are Definitions or Lemmas. The numbering and labels should be harmonized.
  2. [Table 1] The notation for q^m is ambiguous: entries like '72' and '112' are hard to read as powers. Also, some listed pairs (q,m) do not satisfy Eq. (11); the authors should clarify whether those rows are direct computational existence checks rather than consequences of Lemma 3.4.
  3. [Definitions 3.3 and 3.5] The definitions of nice point and nice t-bar should explicitly require t-bar ≠ 0. The subsequent claim that t-bar, ζ t-bar, ..., ζ^r t-bar gives (r+1)^2 distinct points depends on this; as written, the case t-bar = 0 is not excluded.
  4. [5, proof of Lemma 5.2] The assertions about the singular fibers (2I_8, 4I_2) and the discriminant are stated without derivation. Since these are used to apply the Shioda-Tate formula, a short computation or a precise reference would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: distance bound is derived from pole divisors and evaluation-set geometry, not from the target d; the claimed sharpness is checked by brute-force tables. The only flagged issue (Lemma 3.4's dropped Chebotarev error term) is a nonemptiness/correctness gap, not a circular reduction.

full rationale

The central derivation chain is not circular. The code's dimension is dim L by injectivity of evaluation (Lemma 4.1), which uses distinct x-coordinates per vertical fiber; locality and availability are proved by Vandermonde interpolation in Lemma 4.2, with the space L (Eq. 18) restricting to degree ≤ r−1 polynomials on each fiber—this is the standard design of AG-LRCs and not a fitted prediction. The distance bound Eq. (33) is obtained by computing the Newton arc of P_t at t=∞ (Lemma 4.3), deriving valuations (Lemma 4.6), and bounding max pole degree (2r^2−2r−1) and a common zero of multiplicity 2 (Eqs. 28, 32); it does not use the claimed d as an input. Sharpness for r=3 is supported by explicit exhaustive/branch examples in Table 1, not by tuning constants to match d. Proposition 4.10's exact distance 8 for b=1 is proved by constructing f_min and proving no L-function can have more zeros in S_t; this is an independent extremal argument. The one self-citation [23] (Salgado–Várilly-Alvarado–Voloch) appears in Section 5 and supports an alternative recovery proof; since the same parameters were already proven in Lemma 4.2 and [23] is a published, parameter-free external theorem, it is not load-bearing and does not make the derivation circular. I note a real non-circular correctness gap: Lemma 3.4 invokes Chebotarev with count q^m/(m|G|)+O(q^{m/2}/m), then drops the error term and concludes positivity from q^m/m ≥ 2(r+1)!; as written the error can dominate, so existence of a nice t and hence nonemptiness of S_B is not rigorously established. This is a gap in an input hypothesis, not a case of the paper predicting its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters; the construction depends on an existence result from Chebotarev and on standard theorems from function fields and elliptic surfaces. The family is chosen ad hoc, but no hidden degrees of freedom are fitted to data.

assumptions (5)
  • standard math Chebotarev Density Theorem for function fields (Rosen, Theorem 9.13B) with the error term O(q^{m/2}/m) being negligible under condition q^m/m ≥ 2(r+1)!
    Used in Lemma 3.4 to guarantee existence of a completely splitting place; the proof ignores the constant in the O term.
  • standard math Newton polygon theorem for Kummer extensions (Beelen's unpublished notes, Theorem 2.5/2.6)
    Used to determine splitting of (t=∞) and valuations of x in F_qm(t,x); not proved in paper.
  • standard math Shioda-Tate formula and [23, Lemmas VI.1, VI.2] for recovery on elliptic curves
    Used in Section 5 to establish the alternative recovery procedure for r=3.
  • domain assumption E_3 is a K3 surface with singular fibers 2I_8 and 4I_2, and the fibration Π_x has MW group Z/2Z
    Asserted in Section 5 from an 'inspection of the discriminant' without showing the computation; underlies the horizontal recovery reinterpretation.
  • ad hoc to paper The family E_r and M_r and the space L are chosen to make the Newton arc have three segments and the Vandermonde recovery work
    The construction is purpose-built; the paper provides no independent justification for this family beyond the resulting code parameters.

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Pith. "Pith review of Locally Recoverable Codes with availability from a family of fibered surfaces." pith.science (2026). https://pith.science/paper/NDWOVIF2

@misc{pith2026251208100,
  author       = {Pith},
  title        = {Pith review of: Locally Recoverable Codes with availability from a family of fibered surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NDWOVIF2}},
  note         = {Machine review of arXiv:2512.08100}
}
abstract

We construct Locally Recoverable Codes (LRCs) with availability $2$ from a family of fibered surfaces. To obtain the locality and availability properties, and to estimate the minimum distance of the codes, we combine techniques coming from the theory of one-variable function fields and from the theory of fibrations on surfaces. When the locality parameter is $r=3$, we obtain a sharp bound on the minimum distance of the codes. In that case, we give a geometric interpretation of our codes in terms of doubly elliptic surfaces. In particular, this provides the first instance of an error correcting code constructed using a (doubly elliptic) K3 surface.

Figures

Figures reproduced from arXiv: 2512.08100 by the authors.

Figure 1
Figure 1. General shape of the Newton arc of a polynomial φ(T) with respect to a place Q of F. Let now OQ be the DVR associated to the place Q. Then, given any φ(T) ∈ F[T], there always exists some element f ∈ F \ {0} such that fφ(T) ∈ OQ[T] and the valuation of at least one coefficient of fφ(T) at Q is zero. The Newton arc ΓQ(fφ(T)) with respect to Q of the polynomial fφ(T) ∈ F[T] obtained in this way is then just a vertical… view at source ↗
Figure 2
Figure 2. Some subfields of Fqm(t, x). With the notations just introduced above, let now E¯ r and M¯ r denote, respectively, the compactification of Er and Mr in P 2 [x:y:z] × P 1 [t:w] . For a point P := [¯x : ¯x r+1 2 + 1 : 1;t¯ : 1] ∈ M¯ r(Fqm) we define two associated subsets of M¯ r(Fqm) HP := {[¯x : ¯x r+1 2 + 1 : 1; ζ i t¯ : 1] ∈ M¯ r(Fqm) | i = 1, . . . , r}, VP := {[x : x r+1 2 + 1 : 1;t¯ : 1] ∈ M¯ r(Fqm) | Pt¯(x) = … view at source ↗
Figure 3
Figure 3. Visual intuition behind the code construction: the picture gives an idea of how to visualise the set St¯ of (r + 1)2 nice points obtained from a nice t¯. Each point lies on exactly one vertical fiber Π−1 t (ζ j t¯), j = 0, . . . , r, and on exactly one horizontal fiber Π−1 x (¯x), where ¯x is one of the distinct r + 1 solutions of Pt¯(x) = 0. In the picture, the vertical fibers are depicted in red, the horizontal on… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Newton arc of P˜ t(T) with respect to (t = ∞). By Theorem 2.5 and Theorem 2.6, there are only two possibilities for the splitting behaviour of (t = ∞) in Fqm(t, x)/Fqm(t), which are described in the following lemma. Lemma 4.4. Let notations be as in Theorem 4.3. Then …
Figure 5
Figure 5. Figure 5: Splitting of (t = ∞) in case (1) of Theorem 4.4 Fqm(t, x) P1 P2 P3 Fqm(t) (t = ∞) r+1 e=1 f=1 e=1 f=1 e= r−1 2 f=2 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Splitting of (t = ∞) in case (2) of Theorem 4.4 Remark 4.5. By the law of quadratic reciprocity, −1 is a square modulo q m if and only if q m ≡ 1 (mod 4). If r = 3, since we are assuming q ≡ 1 (mod r + 1), we are hence always in case (1) of Theorem 4.4. We now compute …

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Cited by 1 Pith paper

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

  1. Locally recoverable codes from elliptic surfaces with availability and hierarchical locality

    cs.IT 2026-05 unverdicted novelty 6.0 of 10

    Constructions of locally recoverable codes from elliptic surfaces that provide availability t>2, hierarchical locality, and their combination via torsion groups and fibered geometry.

Reference graph

Works this paper leans on

32 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Aguilar, A

    K. Aguilar, A. ´Alvarez, R. Ardila, P. S. Ocal, C. R. Avila, A. V´ arilly-Alvarado,Locally recoverable algebro-geometric codes from projective bundles, arXiv preprint arXiv:2409.04201 (2024)

    arXiv 2024

  2. [2]

    Ballentine, A

    S. Ballentine, A. Barg, S. Vl˘ adut ¸,Codes with hierarchical locality from covering maps of curves, IEEE Transactions on Information Theory65.10(2019), 6056–6071

    2019

  3. [3]

    A. Barg, K. Haymaker, E.W. Howe, G.L. Matthews, A. V´ arilly-Alvarado,Locally recoverable codes from algebraic curves and surfaces, Algebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA, February 2016 (2017), 95–127

    2016

  4. [4]

    Beelen,Kummer theory and Newton polygons, unpublished notes, Technical University of Denmark (2023)

    P. Beelen,Kummer theory and Newton polygons, unpublished notes, Technical University of Denmark (2023)

    2023

  5. [5]

    Beelen,Algebraic geometry and coding theory, Phd Thesis 1 (Research TU/e / Graduation TU/e), Mathematics and Computer Science, Technische Universiteit Eindhoven (2001)

    P. Beelen,Algebraic geometry and coding theory, Phd Thesis 1 (Research TU/e / Graduation TU/e), Mathematics and Computer Science, Technische Universiteit Eindhoven (2001)

    2001

  6. [6]

    J. Berg, B. Malmskog, W. West,Codes with Hierarchical Locality on Artin-Schreier Surfaces, arXiv preprint arXiv:2405.19533 (2024)

    arXiv 2024

  7. [7]

    Beelen, R

    P. Beelen, R. Pellikaan,The Newton Polygon of Plane Curves with Many Rational Points, Designs, Codes and Cryptog- raphy21(2000), 41–67

    2000

  8. [8]

    Dimakis, P

    A. Dimakis, P. Godfrey, Y. Wu, M. Wainwright, K. Ramchandran,Network Coding for Distributed Storage Systems Transactions on Information Theory, IEEE56(2010), 4539–4551

    2010

Show all 32 references
  1. [9]

    El Fadil,Prime ideal factorization in a number field via Newton polygons, Czechoslovak Mathematical Journal71.2 (2021), 529–543

    L. El Fadil,Prime ideal factorization in a number field via Newton polygons, Czechoslovak Mathematical Journal71.2 (2021), 529–543

    2021

  2. [10]

    Gopalan, C

    P. Gopalan, C. Huang, H. Simitci, S. YekhaninOn the locality of codeword symbols, IEEE Transactions on Information Theory58.11(2012), 6925-6934

    2012

  3. [11]

    Han, L.A

    J. Han, L.A. Lastras-Montano,Reliable memories with subline accesses, 2007 IEEE International Symposium on Informa- tion Theory (2007), 2531–2535

    2007

  4. [12]

    Huang, M

    C. Huang, M. Chen, J. Lin,Pyramid codes: Flexible schemes to trade space for access efficiency in reliable data storage systems, ACM Transactions on Storage (TOS)9.1(2013), 1–28

    2013

  5. [13]

    Goss,Basic structures of function field arithmetic, Springer Science & Business Media, Berlin, Heidelberg (2012)

    D. Goss,Basic structures of function field arithmetic, Springer Science & Business Media, Berlin, Heidelberg (2012)

    2012

  6. [14]

    26 CEC ´ILIA SALGADO AND LARA VICINO

    V.D.Goppa,A new class of linear correcting codes, Problemy Peredachi Informatsii,6, n.3, (1970), 24–30. 26 CEC ´ILIA SALGADO AND LARA VICINO

    1970

  7. [15]

    V.D.Goppa,A rational representation of codes and (L,g)-codes, Problemy Peredachi Informatsii,7, n.3, (1971), 41–49

    1971

  8. [16]

    V.D.Goppa,Codes associated with divisors, Problemy Peredachi Informatsii,13, n.1, (1977), 33–39

    1977

  9. [17]

    V.D.Goppa,Codes on algebraic curves, Doklady Akademii Nauk SSSR,259, n.6, (1981), 1289–1290

    1981

  10. [18]

    Seriya Matematicheskaya,46, n.4, (1982), 762–781

    V.D.Goppa,Algebraico-geometric codes, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya,46, n.4, (1982), 762–781

    1982

  11. [19]

    Haymaker, B

    K. Haymaker, B. Malmskog, G.L. Matthews,Locally recoverable codes with availabilityt≥2from fiber products of curves, Advances in Mathematics of Communications12, n. 2, (2018), 317–336

    2018

  12. [20]

    Haymaker, B

    K. Haymaker, B. Malmskog, G.L. Matthews,Algebraic hierarchical locally recoverable codes with nested affine subspace recovery, Designs, Codes and Cryptography93.1, (2025), 111–132

    2025

  13. [21]

    Høholdt, J.H

    T. Høholdt, J.H. van Lint, and R. Pellikaan,Algebraic geometry codes, Handbook of coding theory, Vol. I, II, North- Holland, Amsterdam (1998), 871–961

    1998

  14. [22]

    Rosen,Number theory in function fields, Springer Science & Business Media, Berlin, Heidelberg (2013)

    M. Rosen,Number theory in function fields, Springer Science & Business Media, Berlin, Heidelberg (2013)

    2013

  15. [23]

    Salgado, A

    C. Salgado, A. V´ arilly-Alvarado, J. F. Voloch,Locally recoverable codes on surfaces, IEEE Transactions on Information Theory67.9(2021), 5765–5777

    2021

  16. [24]

    Shimada,On elliptic K3 surfacesThe Michigan Mathematical Journal47(3), (2005)

    I. Shimada,On elliptic K3 surfacesThe Michigan Mathematical Journal47(3), (2005)

    2005

  17. [25]

    Schuett, T

    M. Schuett, T. Shioda,Elliptic Surfaces, Adv. Stud. Pure Math., 51–160 (2010)

    2010

  18. [26]

    Silverman,The arithmetic of elliptic curves2nd ed., Graduate Texts in Mathematics, vol.106, Springer, Dordrecht, (2009)

    J. Silverman,The arithmetic of elliptic curves2nd ed., Graduate Texts in Mathematics, vol.106, Springer, Dordrecht, (2009)

    2009

  19. [27]

    Stichtenoth,Algebraic function fields and codes, Springer Graduate Texts in Mathematics, Berlin, Heidelberg (2009)

    H. Stichtenoth,Algebraic function fields and codes, Springer Graduate Texts in Mathematics, Berlin, Heidelberg (2009)

    2009

  20. [28]

    I. Tamo, A. Barg,A family of optimal locally recoverable codes, IEEE Transactions on Information Theory60.8(2014), 4661–4676

    2014

  21. [29]

    I. Tamo, A. Barg,Bounds on locally recoverable codes with multiple recovering sets, 2014 IEEE International Symposium on Information Theory (2014), 691–695

    2014

  22. [30]

    I. Tamo, A. Barg, S.G. Vl˘ adut ¸,Locally recoverable codes on algebraic curvesIEEE Transactions on Information Theory, 63, Issue 8, (2017) 4928–4939

    2017

  23. [31]

    I. Tamo, A. Barg, A. Frolov,Bounds on the parameters of locally recoverable codes, IEEE Transactions on Information Theory62.6(2016), 3070–3083

    2016

  24. [32]

    Tsfasman, S.G

    M.A. Tsfasman, S.G. Vl˘ adut ¸, D. NoginAlgebraic Geometric Codes: Basic Notions, American Mathematical Society, USA (2007)

    2007

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