time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as

[[18q^2,8,2q]]_{q\in \mathbb{N}}

and

[[18q,8,\leq 2q]]_{q\in \mathbb{N}}

respectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with odd

q

. We also show that the

[[90,8,10]]

bivariate bicycle code is a member of the balanced product code family that saturates the distance bound. We also find balanced product codes that saturate the bound up to

q\leq8

through a numerical search.

DOI: 10.48550/arxiv.2411.03302

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\mathcal{O}(d)

[[18q^2,8,2q]]_{q\in \mathbb{N}}

and

[[18q,8,\leq 2q]]_{q\in \mathbb{N}}

q

. We also show that the

[[90,8,10]]

bivariate bicycle code is a member of the balanced product code family that saturates the distance bound. We also find balanced product codes that saturate the bound up to

q\leq8

through a numerical search.

D

pairwise disjoint physical representatives, enabling parallel gauging and a space--time tradeoff for logical measurements. Finally, we construct an addressable surface code interface for injecting externally prepared resource states into selected logical qubits. Together, these results establish La-cross codes as a simple, low-connectivity generalisation of surface code computation, retaining analogous computational primitives while providing a complete Clifford+

T

instruction set across multiple encoded qubits.

[\![n=2^m,k={m \choose m/2}\approx n/\sqrt{π\log_2(n)/2},d=2^{m/2}=\sqrt{n}]\!]

self-dual quantum Reed$\unicode{x2013}$Muller codes, where

m

is a positive even number. For any code in this family, we construct a generating set of the full logical Clifford group comprising only transversal and fold-transversal gates, thus enabling the implementation of any addressable Clifford gate. To our knowledge, this is the first known construction of the full logical Clifford group using only transversal and fold-transversal gates without requiring ancilla qubits for a family of codes in which

k

grows near-linearly in

n