While quantum low-density parity check (qLDPC) codes are a low-overhead means of quantum information storage, it is valuable for quantum codes to possess fault-tolerant features beyond this resource efficiency. Recently, we introduced trivariate tricycle (TT) codes [1]. TT codes are a natural generalisation of the bivariate bicycle codes. They are a family of CSS codes based on a length-3 chain complex, and are defined from three trivariate polynomials, with the 3D toric code (3DTC) belonging to this construction. 
In my talk I will focus on the rich set of fault tolerant gates these codes support, including a large set of transversal Clifford gates and automorphisms within and between code blocks, and (for several sub-constructions) constant-depth implementations of a (non-Clifford) gate. [1] A. Jacob, C. McLauchlan, D. E. Browne, arXiv:2508.08191 [quant-ph] (2025)