Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$), a $4\ell m$-element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ($7.4\%$ below benchmark) while decoding no worse. At $n=72$, $[[72,14,8]]$ attains $kd^2/n=12.44$---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

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