Recently, a class of quantum secret sharing schemes called communication efficient quantum threshold secret sharing schemes (CE-QTS) was introduced. These schemes reduced the communication cost during secret recovery. In this paper, we introduce a general class of communication efficient quantum secret sharing schemes (CE-QSS) which include both threshold and non-threshold schemes. We propose a framework for constructing CE-QSS schemes to generalize the earlier construction of CE-QTS schemes which was based on the staircase codes. The main component in this framework is a class of quantum codes which we call the extended Calderbank-Shor-Steane codes. These extended CSS codes could have other applications. We derive a bound on communication cost for CE-QSS schemes. Finally, we provide a construction of CE-QSS schemes meeting this bound using the proposed framework.

In the above article, as one of the results, we proposed a construction for universal CE-QTS schemes. However, we recently found an error in this construction. We found that the Vandermonde matrix used for the encoding in this construction leads to a failure in secret recovery. Here we shortly describe why this error occurs and provide a rectification. By replacing the Vandermonde matrix with a Cauchy matrix, the secret recovery in the construction goes through. However, the construction now needs a higher field size. With this rectification, the construction is now correct. The remaining results in the article remain unaffected.

A$((k,n))$quantum threshold secret sharing (QTS) scheme is a quantum cryptographic protocol for sharing a quantum secret among$n$parties such that the secret can be recovered by any$k$or more parties while$k-1$or fewer parties have no information about the secret. Despite extensive research on these schemes, there has been very little study on optimizing the quantum communication cost during recovery. Recently, we initiated the study of communication efficient quantum threshold secret sharing (CE-QTS) schemes. These schemes reduce the communication complexity in QTS schemes by accessing$d>k$parties for recovery; here$d$is fixed ahead of encoding the secret. In contrast to the standard QTS schemes which require$k$qudits for recovering each qudit in the secret, these schemes have a lower communication cost of$\frac {d}{d-k+1}$. In this paper, we further develop the theory of communication efficient quantum threshold schemes. Here, we propose universal CE-QTS schemes which reduce the communication cost for all$d>k$simultaneously. We provide a framework based on ramp quantum secret sharing to construct CE-QTS and universal CE-QTS schemes. We give another construction for universal CE-QTS schemes based on Staircase codes. We derived a lower bound on communication complexity and show that our constructions are optimal. Finally, an information theoretic model is developed to analyse CE-QTS schemes and the lower bound on communication complexity is proved again using this model.

Quantum secret sharing (QSS) is a cryptographic protocol in which a quantum secret is distributed among a number of parties where some subsets of the parties are able to recover the secret while some subsets are unable to recover the secret. In the standard ((k, n)) quantum threshold secret sharing scheme, any subset of k or more parties out of the total n parties can recover the secret while other subsets have no information about the secret. But recovery of the secret incurs a communication cost of at least k qudits for every qudit in the secret. Recently, a class of communication efficient QSS schemes were proposed which can improve this communication cost to $\frac{d}{{d - k + 1}}$ by contacting d ≥ k parties where d is fixed prior to the distribution of shares. In this paper, we propose a more general class of ((k, n)) quantum secret sharing schemes with low communication complexity. In these schemes the combiner can contact any d parties at the time of recovery where k ≤ d ≤ n. This is the first such class of universal communication efficient quantum threshold schemes.

The characterization of the storage-repair bandwidth tradeoff of (n, k, d)-regenerating codes under the exact-repair setting remains an open problem. The problem has been solved only for the special case of (n, k, d) = (4, 3, 3). In the present paper, we characterize the tradeoff for the larger family of parameters (n, k = 3, d = n - 1). This is accomplished by constructing an (n, k <; d, d)-regenerating code, referred to as the improved layered code. In the case when (n, k = 3, d = n - 1), the code operates on a point that coincides with an interior point of a recently derived outer bound on the tradeoff. The code also achieves an interior point on the outer bound for the parameter set (n, k = 4, d = n - 1).

While the tradeoff between the amount of data stored and the repair bandwidth of an (n, k, d) regenerating code has been characterized under functional repair (FR), the case of exact repair (ER) remains unresolved. It is known that there do not exist ER codes which lie on the FR tradeoff at most of the points. The question as to whether one can asymptotically approach the FR tradeoff was settled recently by Tian who showed that in the (4, 3, 3) case, the ER region is bounded away from the FR region. The FR tradeoff serves as a trivial outer bound on the ER tradeoff. In this paper, we extend Tian's results by establishing an improved outer bound on the ER tradeoff which shows that the ER region is bounded away from the FR region, for any (n, k, d). Our approach is analytical and builds upon the framework introduced earlier by Shah et. al. Interestingly, a recently-constructed, layered regenerating code is shown to achieve a point on this outer bound for the (5, 4, 4) case. This represents the first-known instance of an optimal ER code that does not correspond to a point on the FR tradeoff.