We study the problem of constructing balanced generator matrices for Reed-Solomon and Tamo-Barg codes. More specifically, we are interested in realizing generator matrices, for the full-length cyclic versions of these codes, where all rows have the same weight and the difference in weight between any columns is at most one. The results presented in this paper translate to computationally balanced encoding schemes, which can be appealing in distributed storage applications. Indeed, the balancedness of these generator matrices guarantees that the computation effort exerted by any storage node is essentially the same. In general, the framework presented can accommodate various values for the required row weight. We emphasize the possibility of constructing sparsest and balanced generator matrices for Reed-Solomon codes, i.e., each row is a minimum distance codeword. The number of storage nodes contacted once a message symbol is updated decreases with the row weight, so sparse constructions are appealing in that context. Results of similar flavor are presented for cyclic Tamo-Barg codes. In particular, we show that for a code with minimum distance d and locality r , a construction in which every row is of weight d + r - 1 is possible. The constructions presented are deterministic and operate over the codes' original underlying finite field. As a result, efficient decoding from both errors and erasures is possible thanks to the plethora of efficient decoders available for the codes considered.
Halbawi, W., Liu, Z., Duursma, I. M., Dau, H., & Hassibi, B. (2019). Sparse and Balanced Reed–Solomon and Tamo–Barg Codes. IEEE Transactions on Information Theory, 65(1), 118–130. doi:10.1109/tit.2018.2873128
H. Dau was supported in part by NSF under Grant CCF 1526875 and in part by the Center for Science of Information under Grant NSF 0939370. B. Hassibi was supported in part by the National Science Foundation under Grant CNS-0932428, Grant CCF-1018927, Grant CCF1423663, and Grant CCF-1409204, in part by a grant from Qualcomm Inc., in part by the NASA’s Jet Propulsion Laboratory through the President and Directors Fund, in part by King Abdulaziz University, and in part by the King Abdullah University of Science and Technology. This paper was presented in part at the 2016 Information Theory Workshop and the 2017 International Symposium on Information Theory.