Codes Over Absorption Channels

In this paper, we present a novel communication channel, called the absorption channel, inspired by information transmission in neurons. Our motivation comes from in-vivo nano-machines, emerging medical applications, and brain-machine interfaces that communicate over the nervous system. For any given finite alphabet, we give codes that can correct absorption errors. For the binary alphabet, we show that the known binary (multiple-)deletion correcting codes already provide a good solution. For a single-absorption error, we show that the Varshamov-Tenengolts codes provide a near-optimal code in our setting. When the alphabet size
$q$
is at least 3, we construct a single-absorption correcting code whose redundancy is at most
$3\log _{q}(n)+O_{q}(1)$
. Then, based on this code and ideas introduced by Gabrys et al. (2022), we give a second construction of single-absorption correcting codes with redundancy
$\log _{q}(n)+12\log _{q}\log _{q}(n)+O_{q}(1)$
, which is optimal up to an
$O\left ({\log _{q}\log _{q}(n)}\right)$
. Here,
$O_{q}(1)$
denotes a number dependent on
$q$
but independent of the code-length
$n$
. Finally, we apply the syndrome compression technique with pre-coding to obtain a subcode of the single-absorption correcting code. This subcode can combat multiple absorption errors and has low redundancy. For each setup, efficient encoders and decoders are provided.