A Kraft-Type Sufficient Condition for the Existence of $D$-Ary Fix-Free Codes

A greedy scheme called Greedy Codeword Assignment Scheme (GCAS) is proposed to assign D-ary codewords to the given code-lengths ¿
₁
,¿
₂
,...,¿
_n
, so that they satisfy the fix-free property. This scheme guarantees that a D-ary fix-free code can be obtained whenever ¿
_i=1

ⁿ
D
^-¿
¿ ¿(D), where ¿(D) is equal to 5/8 for D even and very close to 5/8 for D odd. This result can be regarded as an extension of Yekhanin's theorem on the existence of binary fix-free codes. In the special case D=2 , the greediness of GCAS enables us to prove that if mini ¿
_i
= 2, the inequality ¿
_i=1

ⁿ
2
<sup>-¿

i</sup>
¿ 21/32 implies the existence of a binary fix-free code with code-lengths ¿
₁
,¿
₂
,...,¿
_n
.