Optimal redundancy for function-correcting symbol-pair codes

Determine the optimal redundancy r_p^f(k,t) required for function-correcting symbol-pair codes (FCSPCs) that recover the value of an arbitrary function f: {0,1}^k Im(f) in the presence of up to t errors under the symbol-pair distance.

Background

The paper extends the function-correcting-code framework from Hamming distance to symbol-pair distance, which measures discrepancies between consecutive symbol pairs. An FCSPC encodes a binary message u together with r redundancy symbols and must ensure pairwise distance at least 2t+1 whenever two messages have different function values.

For FCSPCs, the paper defines r_p^f(k,t) as the smallest redundancy for which such an encoding exists. Prior work provides upper and lower bounds for the Hamming weight function, and the paper improves the lower bound for that particular function, but the optimal redundancy in the general FCSPC setting remains undetermined.