A Bound for Error-Correcting Codes

We consider the class of secret sharing schemes where there is no a priori bound on the number of players n but where each of the n share-spaces has fixed cardinality q. We show two fundamental lower bounds on the threshold gap of such schemes. The threshold gap g is defined as r − t, where r is minimal and t is maximal such that the following holds: for a secret with arbitrary a priori distribution, each r-subset of players can reconstruct this secret from their joint shares without error (r-reconstruction) and the information gain about the secret is nil for each t-subset of players jointly (t-privacy). Our first bound, which is completely general, implies that if 1 ≤ t < r ≤ n, then g ≥ n−t+1 q independently of the cardinality of the secret-space. Our second bound pertains to Fq-linear schemes with secret-space F k q (k ≥ 2). It improves the first bound when k is large enough. Concretely, it implies that g ≥ n−t+1 q + f (q, k, t, n), for some function f that is strictly positive when k is large enough. Moreover, also in the Fq-linear case, bounds on the threshold gap independent of t or r are obtained by additionally employing a dualization argument. As an application of our results, we answer an open question about the asymptotics of arithmetic secret sharing schemes and prove that the asymptotic optimal corruption tolerance rate is strictly smaller than 1.

Section: Related Workmentioning

confidence: 99%

Bounds on the Threshold Gap in Secret Sharing and its Applications

Cascudo

Cramer

Xing

2013

“…If the minimum distance of C 1 is 8 then all codewords of C 1 except the zero and the all-ones vectors have weight 8 and C is Type IV-II code. Hence, the minimum distance of C1 is 4 and its dimension k1 is at least 2.…”

Section: Type IV Codes Of Length 16mentioning

confidence: 99%

Some results on type IV codes over Z4

Bouyuklieva

2002

Abstract-Dougherty, Gaborit, Harada, Munemasa, and Solé have previously given an upper bound on the minimum Lee weight of a Type IV self-dual -code, using a similar bound for the minimum distance of binary doubly even self-dual codes. We improve their bound, finding that the minimum Lee weight of a Type IV self-dual -code of length is at most 4 12 , except when = 4, and = 8 when the bound is 4, and = 16 when the bound is 8. We prove that the extremal binary doubly even self-dual codes of length 24= 32 are not -linear. We classify Type IV-I codes of length 16. We prove that all Type IV codes of length 24 have minimum Lee weight 4 and minimum Hamming weight 2, and the Euclidean-optimal Type IV-I codes of this length have minimum Euclidean weight 8.

“…Let nq(k; d) be the length of a shortest linear code over GF (q) of dimension k and distance d. An important result which is used in our analysis is the Griesmer bound [5], [7] n q (k; d) k01 i=0 dd=q i e: …”

Section: Guaranteed Error Correction Ratementioning

confidence: 99%

Guaranteed error correction rate for a simple concatenated coding scheme with single-trial decoding

Weber

Abdel-Ghaffar²

2000

Abstract-We consider a concatenated coding scheme using a single inner code, a single outer code, and a fixed single-trial decoding strategy that maximizes the number of errors guaranteed to be corrected in a concatenated codeword. For this scheme, we investigate whether maximizing the guaranteed error correction rate, i.e., the number of correctable errors per transmitted symbol, necessitates pushing the code rate to zero. We show that this is not always the case for a given inner or outer code. Furthermore, to maximize the guaranteed error correction rate over all inner and outer codes of fixed dimensions and alphabets, the code rate of one (but not both) of these two codes should be pushed to zero.