Reduced Dimensional Optimal Vector Linear Index Codes for Index Coding Problems with Symmetric Neighboring and Consecutive Side-information

A single unicast index coding problem (SUICP) with symmetric neighboring and consecutive side-information (SNCS) has K messages and K receivers, the kth receiver R
_k
wanting the kth message x
_k
and having the side-information K
_k
= {x
_k-U
,..., x
_k-2
, x
_k-1
}U{x
_k+1
, x
_k+2
,..., x
_k+D
}. Maleki, Cadambe and Jafar obtained the symmetric capacity of this single unicast index coding problem with symmetric neighboring and consecutive side-information (SUICP(SNCS)) and proposed optimal length codes by using Vandermonde matrices. SUICP(SNCS) with arbitrary K, D and U is the only known setting in index coding for which symmetric capacity is known and vector linear index codes are required to achieve capacity. In our earlier work, we gave optimal length (U + 1)-dimensional vector linear index codes for SUICP(SNCS) satisfying some conditions on K, D and U. In this paper, for SUICP(SNCS) with arbitrary K, D and U, we construct optimal length gcd(U+1/K,D-U,U+1)-dimensional vector linear index codes. We prove that the constructed vector linear index code is of minimal dimension if gcd(K - D + U, U + 1) is equal to gcd(K, D - U, U + 1). The proposed construction gives optimal length scalar linear index codes for the SUICP(SNCS) if (U + 1) divides both K and D - U. The proposed construction is independent of field size and works over every field. We give encoding matrices and a low-complexity decoding for the proposed construction.