Sidon Sets on Hypercubes

• Sidon sets in F2^t are defined as subsets where every nontrivial pairwise sum is unique, ensuring that no four distinct elements sum to zero.
• Recent advances have refined upper and lower bounds and achieved record sizes (e.g., size 192 in F2^15) through constructions linked with APN functions.
• These sets are pivotal in constructing binary linear codes with high minimum distance, thereby interconnecting additive combinatorics, cryptography, and coding theory.

A Sidon set in the binary hypercube $\mathbb{F}_2^t$ is a subset with the property that no sum of four distinct elements yields zero, or equivalently, that every nontrivial sum of two distinct elements is unique among unordered pairs. These combinatorial structures are closely linked to classical problems in both additive combinatorics and coding theory, exhibiting deep connections to objects such as almost perfect nonlinear (APN) functions, binary linear codes with high minimum distance, and algebraic curves over finite fields. Recent advances have refined upper and lower bounds on their maximal size and provided new constructions achieving unprecedented parameters.

1. Formal Definition and Structural Properties

Let $\mathbb{F}_2^t$ be the %%%%2%%%%-dimensional vector space over the finite field with two elements, viewed as the vertex set of the $t$-dimensional Hamming hypercube. A subset $M\subseteq\mathbb{F}_2^t$ is defined as a Sidon set if for all four pairwise distinct elements $a, b, c, d\in M$, the sum $a+b+c+d\neq 0$. Equivalently, for all unordered distinct pairs $\{m_1, m_2\}\neq\{m_3, m_4\}$, the two-sums $m_1 + m_2 \neq m_3 + m_4$.

This requirement excludes non-degenerate parallelograms (i.e., quadruples with $a+b = c+d$ for distinct $a,b,c,d$) inside $M$. In characteristic 2, several equivalent formulations exist:

• The set of all two-sums $S_2^*(M) = \{m_1 + m_2 : m_1 \neq m_2 \in M\}$ has cardinality $\binom{|M|}{2}$.
• The set $$S_3^*(M) = \{m_1 + m_2 + m_3 : m_1, m_2, m_3 \text{ distinct}\}$$ has empty intersection with $M$.

Sidon sets in this context are sometimes termed sum-free Sidon sets when additionally $M$ itself contains no solution to $m_1 + m_2 = m_3$ with all $m_i$ distinct—this property is essential in their correspondence with codes of large minimum distance (Czerwinski et al., 2023).

2. Classical Bounds and Exact Values

The maximum size $s_{\max}(t)$ of a Sidon set in $\mathbb{F}_2^t$ is constrained by several combinatorial arguments:

• The trivial upper bound arises from observing that the $\binom{|M|}{2}$ distinct nonzero two-sums must be contained in the $2^t - 1$ nonzero vectors, yielding $|M| = O(2^{t/2})$ (Czerwinski et al., 2024).
• Brouwer–Tolhuizen refined these bounds, providing for odd $t$,

$s_{\max}(t) \le 2^{(t+1)/2} - 2$

and for even $t$, a slightly improved bound

$$s_{\max}(t) \le \left\lfloor \sqrt{2^{t+1} + 0.5} \right\rfloor - \lambda_{a,b,\delta}$$

with $\lambda_{a,b,\delta}$ defined in terms of division remainders and parity, always at least 1 for $t\ge 6$ (Czerwinski et al., 2023).

• Constructions using BCH/Goppa codes or multiplicative subgroups provide lower bounds: for $t=2n$, $|M| = 2^n+1$ is achievable, and for $t=2n-1$, $|M| = 2^{n-1} + 2^{n/2}$ (for $n$ even), or $2^{n-1} + 2^{(n-1)/2}$ (for $n$ odd).

For small $t$, exhaustive computer searches have determined exact values: $s_{\max}(3) = 4$, $s_{\max}(4) = 6$, $s_{\max}(5) = 7$, $s_{\max}(6) = 9$, $s_{\max}(7) = 12$, $s_{\max}(8) = 18$ (Czerwinski et al., 2023, Nagy, 2022).

3. APN Functions and Hyperplane Intersections

The correspondence between Sidon sets and the graphs of APN functions is central:

$$F\colon \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n \text{ is APN} \iff G_F = \{(x, F(x)) : x \in \mathbb{F}_2^n\} \subseteq \mathbb{F}_2^{2n} \text{ is a Sidon set of size } 2^n.$$

Here, APN means that for all $a \neq 0$, $b \in \mathbb{F}_2^n$, the equation $F(x) + F(x+a) = b$ has at most two solutions. The linearity of an APN function, measured via its Walsh transform,

$$W_F(a, b) = \sum_{x\in \mathbb{F}_2^n} (-1)^{a\cdot x + b\cdot F(x)}, \qquad \operatorname{lin}(F) = \max_{(a,b)\neq(0,0)} |W_F(a,b)|$$

governs the intersection size of $G_F$ with affine hyperplanes $H\subset\mathbb{F}_2^{2n}$, yielding new Sidon sets

$$M = G_F \cap H \subseteq \mathbb{F}_2^{2n-1},\quad |M| = 2^{n-1} + \frac{\operatorname{lin}(F)}{2}.$$

Quadratic APN functions on even $n$ universally attain $\operatorname{lin}(F) = 2^{n/2+1}$, so the constructed Sidon set in $\mathbb{F}_2^{2n-1}$ has size $2^{n-1} + 2^{n/2}$. Recent sporadic APN functions for $n=8$ have been found with $\operatorname{lin}(F) = 128$, yielding sets of size $192$ in $\mathbb{F}_2^{15}$, surpassing all previous records in this dimension (Czerwinski et al., 2024).

The inverse function $x \mapsto x^{-1}$ (for odd $n \ge 5$) and Dobbertin family of APN functions (for $5 \mid n$) provide further constructions with explicit size formulas.

4. Connections to Binary Linear Codes

A Sidon set $M\subseteq \mathbb{F}_2^t$ of size $m \ge t+1$ and span dimension $t$ yields, by taking the nonzero vectors in $M$ as columns of a parity-check matrix, a binary linear $[m-1,\,m-1-t,\,d]$ code with minimum distance at least $d=5$ (Czerwinski et al., 2023). The sum-free Sidon sets correspond precisely to linear codes of distance $5$ under this construction.

A notable application is the Sidon set of size $192$ in $\mathbb{F}_2^{15}$ producing a $[191, 176, 5]$ code—improving the previous best code length with fifteen check bits and minimum distance $5$ (Czerwinski et al., 2024). Generally, an APN function $F$ with $\operatorname{lin}(F)$ yields a $$[\,2^{n-1} + \frac{1}{2}\operatorname{lin}(F) - 1,\,2^{n-1} + \frac{1}{2}\operatorname{lin}(F) - 2n - 1,\,5\,]$$ code.

The absence of $[n, n-t, 5]$ linear codes for certain $n,t$ values directly yields nonexistence results for Sidon subsets of corresponding size in $\mathbb{F}_2^t$ (Czerwinski et al., 2023).

5. Maximal Sidon Sets: Explicit Forms and Classification

A Sidon set $M$ is maximal if it cannot be enlarged within $\mathbb{F}_2^t$ while preserving the Sidon condition. For dimensions $t \le 6$, all affine types are classified:

• $t=1$: size $2$; $t=2$: size $3$; $t=3$: size $4$; $t=4$: size $6$; $t=5$: size $7$.
• $t=6$: unique size $9$ maximal type and a size $8$ type.

For $t=7$, exhaustive computations show all maximal Sidon sets have size $12$, and for $t=8$ sizes in $\{15,16,18\}$ have been found, with $18$ being maximal (Czerwinski et al., 2023, Nagy, 2022).

The completeness problem—classifying all maximal Sidon sets—remains significant, especially regarding whether conic-based constructions in $AG(2, 2^m)$ are complete, which depends on arithmetic conditions involving $m$.

6. Sidon Sets, t-thin Sidon Structures, and Level Sets

A generalization considers $t$-thin Sidon sets: in $A = \mathrm{GF}(2)^n$, $T \subset A$ is $t$-thin Sidon if for each $a \neq 0$, $|T \cap (T+a)| \le t$. In this setting, the Sidon property coincides with the $2$-thin condition, yielding the sharp classical bound $|M| < \sqrt{2|A|} + \tfrac{1}{2}$ for Sidon sets (Nagy, 2022).

Level sets $S_b = f^{-1}(b)$ of a function $f : V \to V'$ with differential uniformity $\delta_f$ are $\delta_f$-thin Sidon sets; for APN $f$, all level sets are Sidon.

7. Recent Advances, Open Questions, and Applications

The construction in (Czerwinski et al., 2024) notably advances the state-of-the-art by:

• Providing explicit infinite families of Sidon sets of record size in odd dimension hypercubes.
• Achieving a Sidon set of size $192$ in $\mathbb{F}_2^{15}$, leveraging sporadic APN functions of high linearity.
• Improving upper bounds for Sidon sets and for APN Walsh linearity: $\operatorname{lin}(F) \le 2^n-4$ for any APN $F$.
• Translating new Sidon set constructions into binary linear codes with parameters exceeding previously established lengths for fixed distance and check bits.

Ongoing problems include determining the maximal value $s_{\max}(t)$ for $t \ge 9$, elucidating completeness for structured Sidon families, and bridging Sidon set theory with APN function nonlinearity, particularly in relation to the Liu–Mesnager–Chen conjecture on maximal vectorial nonlinearity (Nagy, 2022).

The investigation of Sidon sets in the hypercube thus integrates combinatorics, finite field algebra, coding theory, and cryptography, with current research directed at both structural classification and the attainment of sharp size bounds (Czerwinski et al., 2024, Czerwinski et al., 2023, Nagy, 2022).