pith. sign in

arxiv: 2605.04618 · v1 · submitted 2026-05-06 · 💻 cs.IT · math.IT

Constructions of locally repairable codes via concatenated codes

Hengfeng Jin , Fang-Wei Fu This is my paper

Pith reviewed 2026-05-08 16:54 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords locally repairable codesbinary LRCsconcatenated codesGriesmer-like boundJohnson-like boundoptimal codesweight distributiondistributed storage
0
0 comments X

The pith

Binary locally repairable codes meeting the Griesmer-like bound and an improved Johnson-like bound are constructed by concatenating selected linear codes over F4.

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

The paper presents a construction of binary locally repairable codes by concatenating an inner code with outer linear codes chosen from the field with four elements. This yields explicit families whose parameters attain known upper bounds on length, dimension, and locality, including codes that meet a Griesmer-like bound and perfect LRCs. For the case of locality two with disjoint repair groups, the authors derive a stricter Johnson-like bound than previously known and exhibit codes that achieve it. The weight distributions of the resulting binary codes are completely determined by those of the outer F4 codes.

Core claim

Selecting suitable linear codes over F4 as outer codes in a concatenation produces binary LRCs that meet the Griesmer-like bound, include perfect examples, and for locality r=2 attain a new upper bound on code size that improves on the Johnson-like bound of Ma and Ge.

What carries the argument

Concatenation scheme in which outer linear codes over F4 are chosen so that the resulting binary code has locality r and meets the stated bounds, with weight distribution inherited from the outer code.

If this is right

  • Explicit optimal binary LRCs with fully known weight distributions become available for distributed storage applications.
  • The Griesmer-like bound is attained by infinite families of binary LRCs.
  • An improved upper bound holds for binary LRCs of locality 2 with disjoint repair groups, and it is tight.
  • Perfect binary LRCs exist and can be constructed systematically.

Where Pith is reading between the lines

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

  • The same outer-code selection method may produce good codes for other localities once appropriate F4 codes are identified.
  • Knowledge of the full weight distribution allows direct computation of the codes' performance under erasure or error patterns beyond single local repairs.
  • The approach separates the locality guarantee (from the inner code) from the global distance (from the outer code), suggesting similar separations could be tried with other small alphabets.

Load-bearing premise

Linear codes over F4 with the exact parameters needed for the concatenation to produce binary LRCs that reach the Griesmer-like or improved Johnson-like bounds must exist.

What would settle it

For a concrete outer code over F4 with the dimensions and minimum distance used in one of the constructions, the resulting binary code fails to have locality r or fails to meet the claimed bound on minimum distance.

read the original abstract

In recent years, locally repairable codes (LRCs) have attracted considerable attention owing to their pivotal role in distributed storage systems. Since binary linear locally repairable codes can significantly reduce the complexity of both encoding and decoding processes, the construction of binary LRCs has attracted extensive research interest. In this paper, we construct locally repairable codes via concatenated codes and present a systematic approach to select outer codes to obtain optimal binary LRCs, where the outer codes are linear codes over $\mathbb{F}_4$. The weight distributions of the resulting LRCs are determined by the weight distributions of the selected linear codes over $\mathbb{F}_4$. Furthermore, several classes of optimal binary locally repairable codes are constructed, including binary LRCs meeting the Griesmer-like bound, and binary perfect LRCs. Meanwhile, for the locality $r=2$, we improve the Johnson-like bound for binary LRCs with disjoint local repair groups established by Ma and Ge, and construct explicit LRCs that attain this new bound.

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 manuscript proposes constructing binary linear locally repairable codes (LRCs) via concatenation, using linear codes over F_4 as outer codes. It presents a systematic selection method for outer codes, derives the weight distributions of the resulting binary LRCs from those of the F_4 codes, constructs multiple families of optimal binary LRCs (including those attaining a Griesmer-like bound and perfect LRCs), and for locality r=2 improves the Johnson-like bound on binary LRCs with disjoint repair groups (due to Ma and Ge) while giving explicit constructions that meet the new bound.

Significance. If the constructions and bound derivations hold, the work supplies explicit, parameter-controlled binary LRCs useful for distributed storage, together with a concatenation-based design technique that links outer-code weight enumerators directly to LRC parameters. The explicit attainment of the improved r=2 bound and the Griesmer-like examples are concrete contributions; the weight-distribution results also enable further analysis of the codes.

major comments (2)
  1. [§3] §3 (Construction via concatenation): the locality and minimum-distance claims for the binary LRC rest on the outer F_4 code having a prescribed weight distribution after the inner mapping; the manuscript must exhibit at least one explicit [N,K,D] F_4 code (or a family) whose weight enumerator produces the asserted distance and locality parameters, otherwise the optimality statements are unsupported.
  2. [§5] §5 (Improved Johnson-like bound for r=2): the new bound is derived under the assumption of disjoint local repair groups; the concatenation construction must be shown to preserve disjointness (or the paper must state the precise condition on the outer code that guarantees it), because any overlap introduced by the inner mapping would invalidate the bound comparison.
minor comments (2)
  1. Notation for the inner mapping (binary image of F_4 symbols) is introduced without a numbered definition; add an explicit equation or subsection label so that later weight-distribution calculations can cite it directly.
  2. Table 1 (example parameters) lists several F_4 outer codes but omits the resulting binary LRC parameters (n,k,d,r); adding these columns would make the optimality claims immediately verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address the major comments point by point below and will incorporate the necessary clarifications and additions in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (Construction via concatenation): the locality and minimum-distance claims for the binary LRC rest on the outer F_4 code having a prescribed weight distribution after the inner mapping; the manuscript must exhibit at least one explicit [N,K,D] F_4 code (or a family) whose weight enumerator produces the asserted distance and locality parameters, otherwise the optimality statements are unsupported.

    Authors: We agree that an explicit example would strengthen the presentation. Although the paper provides a systematic selection method for outer codes over F_4 and derives the resulting parameters from their weight distributions, we will add in the revision a concrete example of an [N, K, D] linear code over F_4, including its weight enumerator, to explicitly demonstrate how it yields an optimal binary LRC with the claimed locality and distance. revision: yes

  2. Referee: [§5] §5 (Improved Johnson-like bound for r=2): the new bound is derived under the assumption of disjoint local repair groups; the concatenation construction must be shown to preserve disjointness (or the paper must state the precise condition on the outer code that guarantees it), because any overlap introduced by the inner mapping would invalidate the bound comparison.

    Authors: This is a valid point. The improved bound applies to binary LRCs with disjoint local repair groups. In our concatenation construction, the inner mapping is applied symbol-wise to the outer code symbols, and the local repair groups are determined by the supports in the outer code. Since the mapping is independent for each outer symbol, the disjointness of the repair groups from the outer code is preserved in the binary code. We will include a brief explanation in Section 5 clarifying this preservation under the given construction. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions via F4 concatenation with direct parameter mapping

full rationale

The paper derives binary LRC parameters and weight distributions directly from the chosen outer [N,K,D] codes over F4 via standard concatenation, then verifies optimality by explicit matching to the Griesmer-like bound and the improved Johnson-like bound (the latter obtained by tightening Ma-Ge without self-citation dependency). No equation or claim reduces to a fitted input renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain; existence of suitable outer codes is asserted via systematic selection rather than assumed by construction. The derivation remains self-contained in classical coding theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of linear codes over finite fields and the definition of code concatenation; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard algebraic properties of linear codes over finite fields F4 and binary fields
    Invoked for concatenation preservation of locality and weight distribution calculations.

pith-pipeline@v0.9.0 · 5474 in / 1090 out tokens · 34302 ms · 2026-05-08T16:54:31.985572+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    IEEE Transactions on Information Theory58(11), 6925–6934 (2012)

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

    work page 2012

  2. [2]

    IEEE Transactions on Communications72(1), 50–62 (2024)

    Fetrat Qharabagh, M., Ardakani, M.: A family of binary locally repairable codes for coded distributed computing. IEEE Transactions on Communications72(1), 50–62 (2024)

    work page 2024

  3. [3]

    IEEE Transactions on Information Theory60(8), 4661–4676 (2014)

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

    work page 2014

  4. [4]

    IEEE Transactions on Information Theory66(1), 210–221 (2020)

    Jin, L., Ma, L., Xing, C.: Construction of optimal locally repairable codes via automorphism groups of rational function fields. IEEE Transactions on Information Theory66(1), 210–221 (2020)

    work page 2020

  5. [5]

    IEEE Transactions on Commu- nications70(5), 2895–2908 (2022)

    Hao, J., Zhang, J., Xia, S.-T., Fu, F.-W., Yang, Y.: Constructions and weight distributions of optimal locally repairable codes. IEEE Transactions on Commu- nications70(5), 2895–2908 (2022)

    work page 2022

  6. [6]

    IEEE Transactions on Information Theory65(2), 1048–1053 (2019)

    Luo, Y., Xing, C., Yuan, C.: Optimal locally repairable codes of distance 3 and 4 via cyclic codes. IEEE Transactions on Information Theory65(2), 1048–1053 (2019)

    work page 2019

  7. [7]

    IEEE Transactions on Information Theory 66(4), 2402–2416 (2020)

    Cai, H., Miao, Y., Schwartz, M., Tang, X.: On optimal locally repairable codes with multiple disjoint repair sets. IEEE Transactions on Information Theory 66(4), 2402–2416 (2020)

    work page 2020

  8. [8]

    IEEE Transactions on Information Theory66(12), 7465–7474 (2020)

    Hao, J., Xia, S.-T., Shum, K.W., Chen, B., Fu, F.-W., Yang, Y.: Bounds and constructions of locally repairable codes: Parity-check matrix approach. IEEE Transactions on Information Theory66(12), 7465–7474 (2020)

    work page 2020

  9. [9]

    IEEE Transactions on Information Theory62(11), 6268–6283 (2016) 19

    Huang, P., Yaakobi, E., Uchikawa, H., Siegel, P.H.: Binary linear locally repairable codes. IEEE Transactions on Information Theory62(11), 6268–6283 (2016) 19

    work page 2016

  10. [10]

    IEEE Transactions on Communications64(8), 3182–3193 (2016)

    Shahabinejad, M., Khabbazian, M., Ardakani, M.: A class of binary locally repairable codes. IEEE Transactions on Communications64(8), 3182–3193 (2016)

    work page 2016

  11. [11]

    IEEE Transactions on Communications 69(8), 4987–4997 (2021)

    Luo, G., Cao, X.: Constructions of optimal binary locally recoverable codes via a general construction of linear codes. IEEE Transactions on Communications 69(8), 4987–4997 (2021)

    work page 2021

  12. [12]

    IEEE Transactions on Information Theory65(7), 4167–4179 (2019)

    Wang, A., Zhang, Z., Lin, D.: Bounds for binary linear locally repairable codes via a sphere-packing approach. IEEE Transactions on Information Theory65(7), 4167–4179 (2019)

    work page 2019

  13. [13]

    SIAM Journal on Discrete Mathematics33(4), 2509–2529 (2019)

    Ma, J., Ge, G.: Optimal binary linear locally repairable codes with disjoint repair groups. SIAM Journal on Discrete Mathematics33(4), 2509–2529 (2019)

    work page 2019

  14. [14]

    IEEE Transactions on Information Theory61(11), 5787–5794 (2015)

    Cadambe, V.R., Mazumdar, A.: Bounds on the size of locally recoverable codes. IEEE Transactions on Information Theory61(11), 5787–5794 (2015)

    work page 2015

  15. [15]

    In: 2025 IEEE International Symposium on Information Theory (ISIT), pp

    Gao, Y., Fang, W.: Griesmer-optimal locally repairable codes via lengthening Reed-Muller codes. In: 2025 IEEE International Symposium on Information Theory (ISIT), pp. 1–6 (2025)

    work page 2025

  16. [16]

    Designs, Codes and Cryptography91(4), 1209–1232 (2023)

    Fang, W., Chen, B., Xia, S.-T., Fu, F.-W., Chen, X.: Perfect LRCs and k-optimal LRCs. Designs, Codes and Cryptography91(4), 1209–1232 (2023)

    work page 2023

  17. [17]

    Finite Fields and Their Applications91, 102273 (2023)

    Fang, W., Fu, F.-W., Chen, B., Xia, S.-T.: Singleton-optimal LRCs and perfect LRCs via cyclic and constacyclic codes. Finite Fields and Their Applications91, 102273 (2023)

    work page 2023

  18. [18]

    Cambridge University Press, New York (2003)

    Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Code. Cambridge University Press, New York (2003)

    work page 2003

  19. [19]

    IEEE Transactions on Information Theory65(10), 6194–6203 (2019)

    Shen, L.-Z., Fu, F.-W.: The decoding error probability of linear codes over the erasure channel. IEEE Transactions on Information Theory65(10), 6194–6203 (2019)

    work page 2019

  20. [20]

    MIT Press, Cambridge, MA, USA (1965)

    Forney, G.D.: Concatenated Codes. MIT Press, Cambridge, MA, USA (1965)

    work page 1965

  21. [21]

    Cambridge University Press, New York (2006)

    Roth, R.: Introduction to Coding Theory. Cambridge University Press, New York (2006)

    work page 2006

  22. [22]

    Online available at http://www.codetables.de

    Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de. Accessed on 2026-01-07. (2007)

    work page 2026

  23. [23]

    The Electronic Journal of Combinatorics5(1), 1–23 (1998)

    Dodunekov, S., Simonis, J.: Codes and projective multisets. The Electronic Journal of Combinatorics5(1), 1–23 (1998)

    work page 1998

  24. [24]

    20 IEEE Transactions on Information Theory21(1), 106–110 (1975)

    Patel, A.: Maximalq-nary linear codes with large minimum distance (corresp.). 20 IEEE Transactions on Information Theory21(1), 106–110 (1975)

    work page 1975

  25. [25]

    Information and Control8(2), 170–179 (1965)

    Solomon, G., Stiffler, J.J.: Algebraically punctured cyclic codes. Information and Control8(2), 170–179 (1965)

    work page 1965

  26. [26]

    Elsevier, Murray Hill, NJ 07974, U.S

    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier, Murray Hill, NJ 07974, U.S. (1977)

    work page 1977

  27. [27]

    IEEE Transactions on Information Theory44(1), 328–333 (1998)

    Fu, F.-W., Han Vinck, A.J., Shen, S.-Y.: On the constructions of constant-weight codes. IEEE Transactions on Information Theory44(1), 328–333 (1998)

    work page 1998

  28. [28]

    Finite Fields and Their Applications10(2), 159–167 (2004) 21

    Baartmans, A., Yucas, J.L.: Caps in the projective geometryP G(n,4). Finite Fields and Their Applications10(2), 159–167 (2004) 21

    work page 2004