Optimal $(2,δ)$ Locally Repairable Codes via Punctured Simplex Codes
Abstract: Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, \delta)$-LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a $(2, \delta)$-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.
- P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin, “On the locality of codeword symbols,” IEEE Trans. Inf. Theory, vol. 58, no. 11, pp. 6925–6934, 2012.
- N. Prakash, G. M. Kamath, V. Lalitha, and P. V. Kumar, “Optimal linear codes with a local-error-correction property,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2012, pp. 2776–2780.
- L. Jin, “Explicit construction of optimal locally recoverable codes of distance 5 and 6 via binary constant weight codes,” IEEE Trans. Inf. Theory, vol. 65, no. 8, pp. 4658–4663, 2019.
- I. Tamo and A. Barg, “A family of optimal locally recoverable codes,” IEEE Trans. Inf. Theory, vol. 60, no. 8, pp. 4661–4676, 2014.
- G. Luo, M. F. Ezerman, and S. Ling, “Three new constructions of optimal locally repairable codes from matrix-product codes,” IEEE Trans. Inf. Theory, vol. 69, no. 1, pp. 75–85, 2023.
- B. Chen, W. Fang, S.-T. Xia, J. Hao, and F.-W. Fu, “Improved bounds and singleton-optimal constructions of locally repairable codes with minimum distance 5 and 6,” IEEE Trans. Inf. Theory, vol. 67, no. 1, pp. 217–231, 2021.
- W. Fang, B. Chen, S.-T. Xia, and F.-W. Fu, “Singleton-optimal lrcs and perfect lrcs via cyclic codes,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2021, pp. 3261–3266.
- ——, “Bounds and constructions of singleton-optimal locally repairable codes with small localities,” arXiv preprint arXiv:2207.05479, 2022.
- V. R. Cadambe and A. Mazumdar, “Bounds on the size of locally recoverable codes,” IEEE Trans. Inf. Theory, vol. 61, no. 11, pp. 5787–5794, 2015.
- A. S. Rawat, A. Mazumdar, and S. Vishwanath, “Cooperative local repair in distributed storage,” EURASIP Journal on Advances in Signal Processing, vol. 2015, no. 1, pp. 1–17, 2015.
- N. Silberstein and A. Zeh, “Anticode-based locally repairable codes with high availability,” Des. Codes Cryptogr., vol. 86, no. 2, pp. 419–445, 2018.
- ——, “Optimal binary locally repairable codes via anticodes,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2015, pp. 1247–1251.
- G. Luo and X. Cao, “Constructions of optimal binary locally recoverable codes via a general construction of linear codes,” IEEE Trans. Commun., vol. 69, no. 8, pp. 4987–4997, 2021.
- J. Y. Hyun, J. Lee, and Y. Lee, “Infinite families of optimal linear codes constructed from simplicial complexes,” IEEE Trans. Inf. Theory, vol. 66, no. 11, pp. 6762–6773, 2020.
- P. Tan, C. Fan, C. Ding, C. Tang, and Z. Zhou, “The minimum locality of linear codes,” Des. Codes Cryptogr., pp. 1–32, 2022.
- C. Ding and H. Niederreiter, “Cyclotomic linear codes of order 3333,” IEEE Trans. Inf. Theory, vol. 53, no. 6, pp. 2274–2277, 2007.
- K. Ding and C. Ding, “A class of two-weight and three-weight codes and their applications in secret sharing,” IEEE Trans. Inf. Theory, vol. 61, no. 11, pp. 5835–5842, 2015.
- G. Luo and S. Ling, “Application of optimal p𝑝pitalic_p-ary linear codes to alphabet-optimal locally repairable codes,” Des. Codes Cryptogr., vol. 90, no. 5, pp. 1271–1287, 2022.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.