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A new family of universal self-synchronizable variable-length codes is introduced. This family is not a generalization or improvement of the existing prefix codes, but is based on a new method of integer representation in a mixed base using the radix-2 and the auxiliary radix-3. Upper length bounds for such codes are obtained. The asymptotic estimates for the (2,3)-encoding including the pointwise redundancy are also given. In particular, this implies that the average length of a (2,3)-codeword is shorter than that of Fibonacci code. Elias gamma and delta codes are adopted for the (2,3) -variant with asymptotically shorter codewords as against the original case. Improvement of gamma and delta encoding for all numbers is also presented. One of (2,3)-codes with high density is highlighted as a possible candidate for practical use in data compression. The (2,3) -codes are very simple to construct and they have evident features of strong robustness.