Proposal for encoding the Old Turkic script in the SMP of the UCS

This is a proposal to encode the Old Turkic script in the international character encoding standard Unicode. The script was published in Unicode Standard version 5.2 in October 2009. Old Turkic appears in stone inscriptions from the early 8c CE near the Orkhon River in Mongolia, and a slightly different version is found near the Yenisei River in Siberia in the later 8c CE. By the 9c CE, the script had been replaced by the Uyghur script

Encoding Arguments

Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is effective, but the underlying probabilistic machinery can be daunting. "Encoding arguments" provide an alternative presentation in which probabilistic reasoning is encapsulated in a "uniform encoding lemma". This lemma provides an upper bound on the probability of an event using the fact that a uniformly random choice from a set of size $n$ cannot be encoded with fewer than $\log_2 n$ bits on average. With the lemma, the argument reduces to devising an encoding where bad objects have short codewords. In this expository article, we describe the basic method and provide a simple tutorial on how to use it. After that, we survey many applications to classic problems from discrete mathematics and computer science. We also give a generalization for the case of non-uniform distributions, as well as a rigorous justification for the use of non-integer codeword lengths in encoding arguments. These latter two results allow encoding arguments to be applied more widely and to produce tighter results.Comment: 50 pages, 7 figure

Coding with Encoding Uncertainty

We study the channel coding problem when errors and uncertainty occur in the encoding process. For simplicity we assume the channel between the encoder and the decoder is perfect. Focusing on linear block codes, we model the encoding uncertainty as erasures on the edges in the factor graph of the encoder generator matrix. We first take a worst-case approach and find the maximum tolerable number of erasures for perfect error correction. Next, we take a probabilistic approach and derive a sufficient condition on the rate of a set of codes, such that decoding error probability vanishes as blocklength tends to infinity. In both scenarios, due to the inherent asymmetry of the problem, we derive the results from first principles, which indicates that robustness to encoding errors requires new properties of codes different from classical properties.Comment: 12 pages; a shorter version of this work will appear in the proceedings of ISIT 201

Blind encoding into qudits

We consider the problem of encoding classical information into unknown qudit states belonging to any basis, of a maximal set of mutually unbiased bases, by one party and then decoding by another party who has perfect knowledge of the basis. Working with qudits of prime dimensions, we point out a no-go theorem that forbids shift operations on arbitrary unknown states. We then provide the necessary conditions for reliable encoding/decoding.Comment: To appear in Physics Letters