  We study the communication efficient secret sharing (CESS) problem introduced
by Huang, Langberg, Kliewer and Bruck. A classical threshold secret sharing
scheme randomly encodes a secret into $n$ shares given to $n$ parties, such
that any set of at least $t$, $t<n$, parties can reconstruct the secret, and
any set of at most $z$, $z<t$, parties cannot obtain any information about the
secret. Recently, Huang et al. characterized the achievable minimum
communication overhead (CO) necessary for a legitimate user to decode the
secret when contacting $d\geq t$ parties and presented explicit code
constructions achieving minimum CO for $d=n$. The intuition behind the possible
savings on CO is that the user is only interested in decoding the secret and
does not have to decode the random keys involved in the encoding process. In
this paper, we introduce a new class of linear CESS codes called Staircase
Codes over any field $GF(q)$, for any prime power $q> n$. We describe two
explicit constructions of Staircase codes that achieve minimum communication
and read overheads respectively for a fixed $d$, and universally for all
possible values of $d, t\leq d\leq n$.
