Zero-Error Capacity of a Class of Timing Channels

We analyze the problem of zero-error communication through timing channels that can be interpreted as discrete-time queues with bounded waiting times. The channel model includes the following assumptions: 1) Time is slotted, 2) at most $N$ "particles" are sent in each time slot, 3) every particle is delayed in the channel for a number of slots chosen randomly from the set $\{0, 1, \ldots, K\}$, and 4) the particles are identical. It is shown that the zero-error capacity of this channel is $\log r$, where $r$ is the unique positive real root of the polynomial $x^{K+1} - x^{K} - N$. Capacity-achieving codes are explicitly constructed, and a linear-time decoding algorithm for these codes devised. In the particular case $N = 1$, $K = 1$, the capacity is equal to $\log \phi$, where $\phi = (1 + \sqrt{5}) / 2$ is the golden ratio, and the constructed codes give another interpretation of the Fibonacci sequence.Comment: 5 pages (double-column), 3 figures. v3: Section IV.1 from v2 is replaced with Remark 1, and Section IV.2 is removed. Accepted for publication in IEEE Transactions on Information Theor

Codes Correcting All Patterns of Tandem-Duplication Errors of Maximum Length 3

The set of all $q$-ary strings that do not contain repeated substrings of length $\leq\ell$ forms a code correcting all patterns of tandem-duplication errors of length $\leq\ell$, when $\ell \in \{1, 2, 3\}$. For $\ell \in \{1, 2\}$, this code is also known to be optimal in terms of asymptotic rate. The purpose of this paper is to demonstrate asymptotic optimality for the case $\ell = 3$ as well, and to give the corresponding characterization of the zero-error capacity of the $(\leq 3)$-tandem-duplication channel. This settles the zero-error problem for $(\leq\ell)$-tandem-duplication channels in all cases where duplication roots of strings are unique.Comment: 5 pages (double-column format