Enhanced Ternary Fibonacci Codes

Abstract

Extending previous work on non-binary Fibonacci codes, a new ternary variant is proposed sharing the main features like robustness against errors and ease of encoding and decoding, while improving the compression efficiency relative to other ternary codes. The improvement is based on an increased density of the codes and also shown empirically on large textual examples. A motivation for d-ary codes, with $d>2$, may be the emergence of future technologies that enable the representation of more than just two values in an atomic storage unit.

Appendix: Encoding and decoding

For the encoding, Algorithm 1 assumes that a sequence of integers is given, and an array B is used to temporarily store the trits representing the current integer. The first step is to calculate the length of the representation, including the terminating delimiter 12 or 22, of a given integer x. According to Eq. (2), this will be the smallest $\ell$ for which $x\ge F_{2\ell -3}-1$, so that

$$\ell = \big \lfloor {\textstyle \frac{1}{2}}\log _{\phi }\big (\sqrt{5}(x+1)\big )+3\big \rfloor ,$$

where $\phi =\frac{1+\sqrt{5}}{2}=1.618$ is the golden ratio, and we have used the fact that Fibonacci numbers are given by $F_n=\frac{1}{\sqrt{5}}\phi ^n$, rounded to the nearest integer.

Once the length j is known, the relative index within the block of legal strings of length $j-2$ is iteratively evaluated. Finally, the last two trits are set to 22 or 12 accordingly. Specifically, if $B[r..j-2]=21^{j-2-r}$ with $r\ge 1$ and $j-2-r\ge 0$, then the suffix is 12, otherwise, it is 22.

The decoding procedure of Algorithm 2 works on an array A of trits assumed to contain the concatenated ternary representations of a sequence of integers. It accumulates the value of the current integer in a variable val. A variable status maintains the number of 2-trits, encountered while scanning the current ternary string, that could possibly belong to the suffix of the form $21^s2$, with $s\ge 0$, serving as delimiter. Once the end of the current string has been detected, we know the length of the encoding and val can be updated to account for the shorter strings, according to Eq. (2). Finally, val has to be adjusted because the last two trits, 12 or 22, do not belong to the representation.