Nega–Hadamard Transform Overview

• The nega–Hadamard transform is a spectral method that generalizes the classical Walsh–Hadamard transform by incorporating negaperiodic encoding in Boolean functions.
• It systematically analyzes spectral flatness and autocorrelation zeros, offering insights into negabent properties vital for sequence design and cryptography.
• The transform’s algebraic link via symmetric quadratic functions bridges negabent and bent properties, underpinning complementary sequence and Boolean function analysis.

The nega–Hadamard transform is a spectral transform on Boolean functions arising as a canonical specialization of the root-Hadamard framework. It generalizes classical transforms such as the Walsh–Hadamard and provides a systematic mechanism for encoding negaperiodic spectral and correlation properties of Boolean and generalized Boolean functions. Its significance lies in its direct correspondence with bent and complementary sequences, as well as the algebraic structure underpinning spectral flatness and correlation zeros (Medina et al., 2019).

1. Generalized Boolean Functions and Preliminaries

Let $\mathbb{F}_2^n$ denote the $n$-dimensional vector space over the binary field. A generalized Boolean function is a mapping

$f: \mathbb{F}_2^n \to \mathbb{Z}_q,\quad q \geq 2,$

recovering ordinary Boolean functions for $q=2$. For $f \in GB_n^q$, its complex sign is defined by

$$S_q[f](x) \coloneqq \zeta_q^{f(x)}, \quad \zeta_q = e^{2\pi i/q}.$$

For $q=2$, $S_2[f](x) = (-1)^{f(x)}$. The binary inner product is $u \cdot x = \sum_{i=1}^n u_i x_i \pmod{2}$, and Hamming weight is $wt(x) = \sum_i x_i$.

2. Root-Hadamard Transform and the Nega–Hadamard Case

The root-Hadamard transform defines a broad template embracing several classical transforms and is parametrized as follows:

• Let $q = 2^k$ and $\zeta_q = e^{2\pi i / q}$.
• Let $A = \{a_1, \dots, a_r\}$ be complex roots of unity, with $a_s^q = 1$, and $\{ R_1, \dots, R_r \}$ a partition of coordinates.
• The root-weight is $A_L(x) = \prod_{s=1}^r a_s^{wt(x_{R_s})}$, where $x_{R_s}$ restricts $x$ to $R_s$.

The root-Hadamard transform is defined for $u \in \mathbb{F}_2^n$ by:

$$U_{L, A}[f](u) \coloneqq 2^{-n/2} \sum_{x \in \mathbb{F}_2^n} S_q[f](x) \cdot (-1)^{u \cdot x} \cdot A_L(x).$$

Specializing to $q=2$, $r=1$, $a_1 = -1$, $R_1 = \{1, \dots, n\}$ yields the nega–Hadamard transform, with $A_L(x) = (-1)^{wt(x)}$ and

$$N[f](u) \coloneqq 2^{-n/2} \sum_{x \in \mathbb{F}_2^n} (-1)^{f(x) + u \cdot x + wt(x)}.$$

The classical Walsh–Hadamard transform is recovered for $A_L(x) = 1$.

3. Inversion, Correlation, and Spectral Criteria

The root-Hadamard transform possesses a unitary kernel (up to normalization), admitting explicit inversion. For $y \in \mathbb{F}_2^n$,

$$S_q[f](y) = 2^{n/2} A_L(y) \sum_{u \in \mathbb{F}_2^n} U_{L,A}[f](u) \, (-1)^{y \cdot u}.$$

For the nega–Hadamard specialization:

$$(-1)^{f(y)} = 2^{n/2} (-1)^{wt(y)} \sum_u N[f](u) (-1)^{y \cdot u}.$$

The nega–autocorrelation of $f$ at $z \in \mathbb{F}_2^n$ is

$$C^N_f(z) \coloneqq \sum_{x \in \mathbb{F}_2^n} (-1)^{f(x+z) - f(x)} (-1)^{x \cdot z}.$$

The cross-correlation relation holds:

$$C^N_{f,g}(z) = (-1)^{wt(z)} \sum_{u \in \mathbb{F}_2^n} N[f](u) N[g](u) (-1)^{u \cdot z}.$$

The Parseval identity is $\sum_u |N[f](u)|^2 = 2^n$.

A Boolean function $f \in B_n$ is negabent if $|N[f](u)| = 1$ for all $u$, equivalent to $C^N_f(z) = 0$ for all $z \neq 0$.

4. Spectral Characterization and Relation to Bent Functions

A fundamental property links the nega–Hadamard with the Walsh–Hadamard transform using the symmetric quadratic polynomial $s_2(x) = \sum_{1 \leq i < j \leq n} x_i x_j$. Define $h(x) = f(x) + s_2(x) \pmod{2}$. Then,

$N[f](u) = W[h](u) \quad \forall u,$

where $W$ denotes the Walsh–Hadamard transform.

Consequently, $f$ is negabent if and only if $h$ is bent (when $n$ is even, cf. Parker–Pott 2007). Spectral flatness ($|N[f](u)| \equiv 1$) is equivalent to zero nega-autocorrelation at nonzero shifts.

5. Complementarity and Negaperiodic Correlation

Negaperiodic complementarity manifests via paired spectral sums. Bipolar sequences $a, b \in \{\pm1\}^N$ are a negaperiodic complementary pair if

$C^N_a(k) + C^N_b(k) = 0,\quad k \neq 0.$

Analogously, Boolean functions $f,g \in B_n$ are negacomplementary if $C^N_{f,g}(z) = 0$ for $z \neq 0$.

There is a direct correspondence: for $h_f(x) = f(x) + s_2(x)$ and $h_g(x) = g(x) + s_2(x)$, the functions $f,g$ are negacomplementary if and only if $\{h_f, h_g\}$ forms a Golay complementary pair under the ordinary Walsh–Hadamard framework.

6. Explicit Example

For $n=2$, let $f(x_1, x_2) = x_1 x_2 \pmod{2}$. Then $N[f](u)$ is calculated as:

• $x \in \{(0,0), (0,1), (1,0), (1,1)\}$, with $f(x) = (0,0,0,1)$, $wt(x) = (0,1,1,2)$.
• For $u=(0,0)$: $$N[f](0,0) = \frac{1}{2}\left(1 - 1 - 1 - 1\right) = -1$$.
• For $u=(1,0)$: $N[f](1,0) = \frac{1}{2}(1 - 1 + 1 + 1) = 1$.
• For $u=(0,1)$: $N[f](0,1) = 1$.
• For $u=(1,1)$: $N[f](1,1) = -1$.

Thus $N[f] = (-1, +1, +1, -1)$, which has constant modulus; $f$ is negabent. Inversion recovers $(-1)^{f(y)}$.

7. Context and Significance

The root-Hadamard framework unifies a spectrum of transforms for Boolean and generalized Boolean functions, including Walsh, nega, $2^k$-Hadamard, consta-Hadamard, and $HN$-type transforms. The nega–Hadamard transform corresponds to the case $A = \{-1\}$ over all coordinates. The algebraic connection $f(x) + s_2(x)$ links negabent and bent functions, aligning spectral flatness with autocorrelation zeros. The theory of complementarity—both Golay and negaperiodic—emerges naturally in the context of these transforms, with composition criteria formalized via binary components. This structural unification has foundational significance for the study of Boolean function spectra, sequence design, and combinatorial properties (Medina et al., 2019).