arXiv:2610.22231v1 Announce Type: new Abstract: This paper studies the design of some high-rate, adaptive, low-density parity-check (LDPC) convolutional codes with short constraint length, based on interleaved parity-check polynomials. Starting from some known results regarding the selection of polynomials that yield codes free of short cycles in the associated Tanner graphs, we propose some new constructions based on Golomb rulers, along with relaxed variants that do not strictly satisfy the Golomb ruler condition, but retain similar structural properties. The result is a family of rate-adaptive code constructions with code rates of the form $R=(n_0-1)/n_0$. Conditions for the absence of length-4 cycles are derived, together with upper bounds on the free distance that are met in all considered examples.