Butson Hadamard Codes

• Butson Hadamard Codes are families derived from complex Hadamard matrices that convert phase space orthogonality into strong metric regularity over Z_q.
• They employ logarithmic representations to map roots of unity to residue classes, enabling analysis via covering radius, bentness, and Gray maps.
• Their construction bridges classical coding theory and quantum stabilizer codes, highlighting applications in matrix theory, combinatorics, and cryptography.

Butson Hadamard codes are code families derived from Butson Hadamard matrices, that is, matrices $H\in BH(n,q)$ whose entries are $q$-th roots of unity and satisfy $HH^*=nI_n$. In the current literature, the term usually refers to a code obtained from the logarithmic form $L(H)\in \mathbb Z_q^{n\times n}$ by adjoining all constant shifts of the row set, although closely related conventions replace the row set by the $\mathbb Z_q$-module generated by the rows. Across these variants, the central principle is the same: complex orthogonality in phase space is converted into strong metric regularity for codes over $\mathbb Z_q$, with covering-radius, bentness, Gray-map, propelinear, and morphism phenomena all tied to the underlying matrix structure (Armario et al., 2024, Shi et al., 2023, Armario et al., 2020).

1. Matrix-theoretic basis and logarithmic representation

A Butson Hadamard matrix of order $n$ and phase $q$ is an $n\times n$ complex matrix $H$ whose entries are $q$0-th roots of unity and which satisfies

$q$1

The standard examples are Fourier and character-table matrices: if $q$2 is abelian of order $q$3 and exponent $q$4, then its character table $q$5 lies in $q$6. The literature also uses the dephased convention, under which the first row and first column are all $q$7 multiplicatively, or all $q$8 in logarithmic form, and it works modulo monomial row and column multipliers as the basic equivalence relation (Armario et al., 2024).

The logarithmic representation replaces roots of unity by residue classes. If

$q$9

then

$HH^*=nI_n$0

This finite-ring encoding is the starting point of the code-theoretic theory. It permits the rows of $HH^*=nI_n$1 to be treated as vectors over $HH^*=nI_n$2, while preserving the phase relations responsible for orthogonality. In that sense, $HH^*=nI_n$3 is the bridge between complex Hadamard theory and the algebra of codes over residue rings (Armario et al., 2020).

2. Canonical code constructions and conventions

The literature uses two closely related conventions for the associated code. Both begin with the logarithmic matrix $HH^*=nI_n$4, and both adjoin all constant vectors $HH^*=nI_n$5.

Convention	Definition	Source
Row-set convention	$HH^*=nI_n$6 is the set of rows of $HH^*=nI_n$7; $HH^*=nI_n$8	(Armario et al., 2024)
Generated-code convention	$HH^*=nI_n$9 is the $L(H)\in \mathbb Z_q^{n\times n}$0-code, or $L(H)\in \mathbb Z_q^{n\times n}$1-code, generated by the rows of $L(H)\in \mathbb Z_q^{n\times n}$2; $L(H)\in \mathbb Z_q^{n\times n}$3	(Shi et al., 2023, Armario et al., 2020)

These definitions coincide in emphasizing constant-shift closure. The code is therefore naturally self-complementary in the sense used in later covering-radius arguments. The generated-code convention is especially convenient when BH-codes are compared with additive codes over $L(H)\in \mathbb Z_q^{n\times n}$4, while the row-set convention is more directly tied to the matrix itself.

This code construction is explicitly presented as the natural analogue of the first-order generalized Reed–Muller code. In particular, for $L(H)\in \mathbb Z_q^{n\times n}$5, the code $L(H)\in \mathbb Z_q^{n\times n}$6 records the logarithmic row structure of $L(H)\in \mathbb Z_q^{n\times n}$7 modulo global phase shifts, and when $L(H)\in \mathbb Z_q^{n\times n}$8 with $L(H)\in \mathbb Z_q^{n\times n}$9 even, this recovers the classical bent-function/Reed–Muller paradigm (Armario et al., 2024). The same code may also be viewed geometrically: after embedding $\mathbb Z_q$0 as $\mathbb Z_q$1, the code becomes a spherical code in $\mathbb Z_q$2, so BH-codes admit both finite-ring and Euclidean-spherical interpretations (Shi et al., 2023).

A recurrent misconception is that BH-codes are necessarily linear. The cocyclic examples studied in the propelinear setting show otherwise: a BH-code attached to a cocyclic $\mathbb Z_q$3 matrix is explicitly non-linear, even though it retains strong internal group structure as a full propelinear code (Armario et al., 2020).

3. Metrics, distance, and covering radius

Several inequivalent metrics are used in the theory, and the resulting bounds are metric-sensitive.

Under the Chinese Euclidean metric, for $\mathbb Z_q$4,

$\mathbb Z_q$5

For the corresponding covering radius,

$\mathbb Z_q$6

the existence of a bent sequence yields the lower bound

$\mathbb Z_q$7

while spherical-design arguments give

$\mathbb Z_q$8

for dephased $\mathbb Z_q$9. The Euclidean distance spectrum of the associated spherical code is

$\mathbb Z_q$0

and for $\mathbb Z_q$1 the spherical code meets the Levenshtein bound with parameters $\mathbb Z_q$2 (Shi et al., 2023).

A second framework uses the homogeneous metric on quasi-Frobenius ring alphabets. For $\mathbb Z_q$3, the homogeneous weight is

$\mathbb Z_q$4

If $\mathbb Z_q$5 admits a bent vector, then

$\mathbb Z_q$6

and for $\mathbb Z_q$7,

$\mathbb Z_q$8

On the upper-bound side, strength-$\mathbb Z_q$9 BH-codes satisfy

$n$0

and for self-complementary strength-$n$1 BH-codes over $n$2,

$n$3

These results make explicit that covering radius is controlled by two distinct matrix-derived features: bentness and orthogonal-array strength (Xu et al., 18 Aug 2025).

A third line of work studies codes obtained from normalized $n$4 matrices under homogeneous and non-homogeneous Gray maps. In that setting, the families $n$5 are defined from the rows of the exponent matrix and its translates, and exact minimum distances are proved. In particular, under the non-homogeneous Gray map $n$6, the family $n$7 is a Plotkin-optimal $n$8-ary nonlinear code with parameters

$n$9

and analogous Plotkin-optimal statements are established for $q$0, $q$1, and $q$2 (Acar et al., 2020).

4. Bent vectors, self-duality, and code geometry

Bentness is the principal extremal notion linking BH matrices to BH-codes. For $q$3, a vector $q$4 with entries among the $q$5-th roots of unity is $q$6-bent if

$q$7

where every entry of $q$8 has modulus $q$9. Two special cases are distinguished: $n\times n$0 with $n\times n$1. A more general formulation introduces a multiplier $n\times n$2 and studies self-dual bent sequences satisfying

$n\times n$3

This unifies eigenvector-type and Galois-twisted self-duality conditions (Armario et al., 2024, Shi et al., 2023).

The arithmetic of bent vectors is highly constrained. Using cyclotomic fields, prime ideal factorization, ramification, and self-conjugacy modulo $n\times n$4, one obtains non-existence criteria. If $n\times n$5 is $n\times n$6-bent for $n\times n$7, $n\times n$8, $n\times n$9, and $H$0 is self-conjugate modulo $H$1, then the $H$2-part $H$3 must be a square. For small phases there are sharper constraints: if $H$4 and some entry of $H$5 is a cube root of unity, then necessarily

$H$6

and for $H$7 the analogous conclusion is

$H$8

These conditions feed directly into covering-radius lower bounds, because bent vectors provide explicit vectors that are far from the BH-code (Armario et al., 2024).

Constructive bent-vector theory is equally rich. Tensor products preserve bentness: if $H$9 is conjugate self-dual $q$00-bent and $q$01 is conjugate self-dual $q$02-bent, then $q$03 is conjugate self-dual $q$04-bent. Bush-type Butson Hadamard matrices yield large explicit families: for $q$05 of Bush type there are at least $q$06 self-dual bent vectors and at least $q$07 conjugate self-dual $q$08-bent vectors, while for odd $q$09 a related construction yields $q$10 matrices in $q$11, each admitting self-dual and conjugate self-dual bent vectors (Armario et al., 2024).

The computational study of self-dual bent sequences complements these theoretical constructions. For various $q$12 and lengths $q$13, explicit examples are obtained by solving

$q$14

via Gröbner bases and via eigenspace computations derived from repeated application of the multiplier $q$15. This computational layer is important because it identifies concrete BH-codes whose covering radii achieve or approach the known lower bounds (Shi et al., 2023).

5. Gray maps, propelinear structure, and morphisms

A major algebraic theme is that BH-codes behave well under generalized Gray maps. For $q$16 prime and $q$17, the map

$q$18

is defined from the logarithmic Fourier matrix and is an isometric embedding

$q$19

If $q$20, then the entrywise Gray image $q$21 lies in

$q$22

and the image of a BH-code over $q$23 is again a BH-code over $q$24; for $q$25 this is a binary Hadamard code. More generally, for $q$26 with $q$27, the map $q$28 yields

$q$29

and repeated application produces a matrix

$q$30

whenever $q$31 and every prime dividing $q$32 also divides $q$33 (Armario et al., 2020).

This Gray-map formalism is closely tied to additive-code theory. A $q$34-additive code is isomorphic as a group to a BH-code over $q$35, and its Gray image is a BH-code over $q$36. In that sense, BH-codes are not merely analogous to additive codes over residue rings; they subsume those additive structures inside a phase-space framework (Armario et al., 2020).

Cocyclic matrices endow BH-codes with stronger internal symmetry. If $q$37 is cocyclic, then the associated code $q$38 is a Butson full propelinear code, or BHFP-code. Conversely, in the row-and-column balanced setting, a BHFP-code arising from a BH matrix forces the matrix to be cocyclic. The cocyclic $q$39 example shows that full propelinearity and non-linearity can coexist, so the relevant group structure is not exhausted by linearity over $q$40 (Armario et al., 2020).

At the matrix level, several morphism theorems enlarge the supply of BH-codes. If $q$41 and every prime divisor of $q$42 divides $q$43, then from any $q$44 one can construct $q$45 (Cathain et al., 2019). A spectral method yields complete morphisms

$q$46

generalizing Turyn’s classical $q$47 construction (Egan et al., 2018). More generally, the sound-pair framework constructs morphisms $q$48 from compatible spectral conditions on a template matrix $q$49 and entrywise power conditions on $q$50 (Egan et al., 2017). This suggests a systematic mechanism for transferring BH-code constructions across phases and lengths.

6. Existence theory, classification, and broader coding connections

The existence of BH-codes is inseparable from the existence of the underlying matrices. Computer-aided classification has produced exact enumerations for several small parameters: up to monomial equivalence there are exactly $q$51 matrices in $q$52, $q$53 matrices in $q$54, and $q$55 matrices in $q$56. The same study gives a first example of $q$57 and proves the nonexistence of $q$58, $q$59 for $q$60, and $q$61 (Lampio et al., 2017). For cocyclic matrices with odd prime phase $q$62 and $q$63, the classification is complete: there are exactly $q$64 equivalence classes of cocyclic $q$65, exactly $q$66 of cocyclic $q$67, and exactly $q$68 of cocyclic $q$69 (Egan et al., 2015).

The circulant case is particularly rigid. For prime $q$70, any circulant Butson matrix in the relevant prime-order class is equivalent to the Fourier matrix $q$71, while for distinct primes $q$72 there is no circulant Butson matrix in the mixed-prime case $q$73 (Hiranandani et al., 2013). These results matter for BH-codes because circulant and cocyclic matrix models are standard sources of explicit code families.

Recent construction theory continues to enlarge the matrix pool. A positive-definite-function criterion has been used to prove new nonexistence results and to guide the discovery of $q$74, as well as $q$75-circulant examples $q$76, $q$77, $q$78, and $q$79 (Czifra et al., 14 Nov 2025). A separate recursive construction, using Fourier matrices and Latin-square tensors as input, produces $q$80 in general and $q$81 when $q$82 and $q$83 are even (Adda, 22 Apr 2025). From the coding viewpoint, every such existence theorem yields a new phase-space code seed.

BH matrices also enter adjacent coding frameworks. A constructive proof of quantum-code existence uses BH matrices equivalent to multiple Kronecker products of the Fourier matrix of order $q$84 to build $q$85 quantum stabilizer codes from classical linear codes $q$86 and $q$87. In the diagonal case

$q$88

the converse is especially restrictive: if the resulting quantum code is a stabilizer code, then the normalized BH matrix must be equivalent to the $q$89-fold Kronecker product of the Fourier matrix of order $q$90 (Sarac et al., 2024). This does not redefine BH-codes, but it shows that the same matrix mechanisms that govern BH-codes in classical phase space also govern stabilizer-compatible quantum constructions.

Taken together, these results place Butson Hadamard codes at the intersection of complex Hadamard theory, coding theory over finite rings, cyclotomic arithmetic, spherical coding, and quantum-code construction. Their defining feature is not a single metric or a single algebraic model, but the persistence of Hadamard orthogonality under logarithmic encoding, constant shifts, Gray maps, and bent-vector extremality.