This paper presents new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties, which support fold-transversal Clifford gates without overhead. The BB codes are constructed on a 7×7 grid and have parameters [[98, 6, 12]] and [[162, 8, 12]]. These codes feature explicit bases of logical operators and admit certain fold-transversal Clifford gates, which are essential for fault-tolerant quantum computation. The paper also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes. The BB codes are shown to have favorable properties such as purity, principalness, and symmetry, which allow for the construction of logical operators and gates in a natural way. The paper discusses the homological algebra of CSS codes and their relation to chain complexes, and introduces the concept of pure and principal codes. It also describes fold-transversal gates, including swap-type gates, Hadamard-type gates, and phase-type gates, which are essential for fault-tolerant quantum computation. The paper concludes with two extended examples of BB codes, showing how to construct logical operators and fold-transversal gates in detail. The results have potential applications in quantum error correction and fault-tolerant quantum computation.This paper presents new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties, which support fold-transversal Clifford gates without overhead. The BB codes are constructed on a 7×7 grid and have parameters [[98, 6, 12]] and [[162, 8, 12]]. These codes feature explicit bases of logical operators and admit certain fold-transversal Clifford gates, which are essential for fault-tolerant quantum computation. The paper also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes. The BB codes are shown to have favorable properties such as purity, principalness, and symmetry, which allow for the construction of logical operators and gates in a natural way. The paper discusses the homological algebra of CSS codes and their relation to chain complexes, and introduces the concept of pure and principal codes. It also describes fold-transversal gates, including swap-type gates, Hadamard-type gates, and phase-type gates, which are essential for fault-tolerant quantum computation. The paper concludes with two extended examples of BB codes, showing how to construct logical operators and fold-transversal gates in detail. The results have potential applications in quantum error correction and fault-tolerant quantum computation.