Optimal Ternary Cyclic Codes

• Optimal ternary cyclic codes are linear codes over F3 defined by generator polynomials with cyclic shifts, achieving parameters like [3^m-1, 3^m-1-2m, 4].
• They utilize algebraic constructions based on cyclotomic cosets and minimal polynomials to secure optimal minimum Hamming distances and efficient decoding.
• Recent advances include multi-zero constructions and counterexamples that refine optimality criteria and open new research directions in coding theory.

Optimal ternary cyclic codes are an intensively studied class within algebraic coding theory, characterized by their cyclic structure, definition over the finite field $\mathbb{F}_3$, and the attainment of optimal parameter bounds—most notably, a minimum Hamming distance %%%%1%%%% that meets the sphere-packing bound for given length $n=3^m-1$ and dimension $k$. These codes are central for the design of reliable communication and storage systems, offering maximized error correction for a given rate while allowing efficient algebraic encoding and decoding via their cyclic structure.

1. Algebraic Definitions and Parameters

A ternary cyclic code $C$ is a linear $[n,k,d]$ code over $\mathbb{F}_3$ such that any codeword $(c_0, c_1, \ldots, c_{n-1})$ implies $(c_{n-1}, c_0, c_1, \ldots, c_{n-2})$ is also a codeword. Such codes correspond to ideals in the quotient ring $\mathbb{F}_3[x]/(x^n-1)$, and possess a unique monic generator polynomial $g(x)$ dividing $x^n-1$.

A major focus is on families with parameters $[3^m-1,3^m-1-2m,4]$, where $d=4$ is the maximal possible minimum distance (by the sphere-packing bound) for the given length and dimension. The generator polynomial often takes the form $g(x) = m_1(x) m_e(x)$ where $m_i(x)$ is the minimal polynomial over $\mathbb{F}_3$ of $\alpha^i$ for a primitive element $\alpha \in \mathbb{F}_{3^m}$ and suitable exponent $e$ whose $3$-cyclotomic coset has size $m$ and is disjoint from $C_1$ (Ding et al., 2013, Li et al., 2013).

More general constructions involve additional zeros, e.g., codes $C_{(i_1, i_2, \ldots, i_t)}$ with $t = 2$ or $3$ zeros, and the associated dimension is determined by the sum of the sizes of the involved cyclotomic cosets (Wu et al., 2024).

2. Constructions and Explicit Families

Numerous infinite families of optimal ternary cyclic codes have been established:

• Monomial-Based Families: Codes $\mathcal{C}_{(1,e)}$ built from perfect or almost perfect nonlinear (PN/APN) monomials $x^e$ yield optimal parameters when $e$ satisfies certain algebraic properties relative to $m$. Explicit choices include $e = 2$, $e = (3^h+1)/2$ for odd $h$ coprime to $m$, and several APN-inspired exponents (Ding et al., 2013).
• Welch-Type and Generalized Families: Families such as $\mathcal{C}_{(u,v)}$ with $u=(3^m+1)/2$, $v=2 \cdot 3^\ell+1$ (for $m=2\ell+1$) also meet the optimal bound. These exponents are derived from Welch-type and related functions, with algebraic criteria ensuring no nontrivial codeword of weight $<4$ (Fan et al., 2015).
• Two/Three-Zero Codes: Recent advances yield new families such as $\mathcal{C}_{(0,1,e)}$, $\mathcal{C}_{(1,e,s)}$, $\mathcal{C}_{(2,e)}$, and $\mathcal{C}_{(1,e)}$ for explicitly constructed $e$ depending on congruence relations and field characteristics, all with $d=4$ and dimension determined by the sum of coset sizes (Wu et al., 2024). These families are shown inequivalent to previously known classes.
• Parameter Table: Core Families

Family	Code Parameters	Defining Exponent $e$ (examples)
Monomial (Carlet–Ding–Yuan, Ding–Helleseth)	$[3^m-1,3^m-1-2m,4]$	$e=2$, $(3^h+1)/2$, $3h+1$, APN exponents
Welch-type (Fan et al., 2015)	$[3^m-1,3^m-1-2m,4]$	$u=(3^m+1)/2$, $v=2\cdot3^\ell+1$ (odd $m$)
Wu et al. (Wu et al., 2024)	$[3^m-1,3^m-3m-2,4]$, $[3^m-1,3^m-2m-1,4]$	Structured $e$ values dependent on $m$
Two/Three-zero constructions	$[3^m-1,3^m-1-2m,4]$	$e$ from specific congruences and coset conditions

3. Characterization Criteria and Optimality Proofs

A central algebraic tool is the criterion—originally formulated by Ding and Helleseth—that codifies when $\mathcal{C}_{(1,e)}$ is optimal with $d=4$ (Ding et al., 2013, Li et al., 2013):

• $(\mathrm{C}1)$ $e$ is even and $|C_e|=m$;
• $(\mathrm{C}2)$ the equation $(x+1)^e + x^e + 1 = 0$ in $\mathbb{F}_{3^m}$ has only $x=1$ as solution;
• $(\mathrm{C}3)$ the equation $(x+1)^e - x^e - 1 = 0$ has only $x=0$ as solution.

Proofs of minimal distance rely on showing the absence of codewords of lower weight, typically via analysis of these equations. The sphere-packing (Hamming) bound and Griesmer bound both imply $d \leq 4$ for $[n, n-2m]$ ternary codes with $n=3^m-1$, so these constructions achieve the maximum possible (Ding et al., 2013, Li et al., 2013, Han et al., 2019).

4. Advances, Counterexamples, and Infinite Families

Despite extensive progress, the complete classification of all parameter sets $(m,e)$ producing optimal ternary cyclic codes remains open. Ding and Helleseth's list of nine open problems motivated a decade of research, resulting in:

• Complete resolution of several problems and construction of wider families of $e$ via new cyclotomic and congruence methods (Wu et al., 2024, Zheng et al., 3 Nov 2025, He et al., 11 Jun 2025, Bao et al., 18 Jan 2026).
• Discovery of explicit counterexamples to certain open conjectures for particular $(m,e)$ (notably for $e=(3^h+5)/2$, $e=(3^h-5)/2$), demonstrating that the aforementioned criteria may fail for some parameters due to the existence of nontrivial solutions to $(\mathrm{C}2)$ or $(\mathrm{C}3)$ (Bao et al., 18 Jan 2026, He et al., 11 Jun 2025).
• Identification of new sufficient conditions whereby $e$ subject to congruences such as $e(3^h\pm1) \equiv (3^m-a)/2 \bmod(3^m-1)$, with $a\equiv 3 \bmod 4$, guarantees optimal codes; and nonexistence for certain other cases $a \equiv 1 \bmod 4$ (Bao et al., 18 Jan 2026).

5. Dual Codes and Weight Distributions

The duals of some families of optimal ternary cyclic codes are themselves optimal with respect to the sphere-packing bound, usually possessing parameters $[3^m-1,2m, d']$ with $d'=4$ in favorable cases (Ding et al., 2013, Zhou et al., 2013). Explicit determination of weight distributions for both the codes and their duals, while tractable for small $m$ using exponential sums (trace representations, value distributions of Weil sums, cyclotomic arguments), remains a challenge for general $m$.

Notably, three-weight codes (with precisely three nonzero weights) constructed via trace forms admit optimal duals under specified algebraic constraints, with determined weight enumerators matching those in best-known code tables (Ding et al., 2013, Zhou et al., 2013).

6. Inequivalence, Applications, and Recent Developments

Considerable effort is made to prove inequivalence of new constructions to previously known families, which is crucial for both theoretical completeness and practical deployment. This is assured via cyclotomic coset analysis, explicit $3$-adic expansion argumentation, and generator polynomial comparison (Wu et al., 2024, He et al., 11 Jun 2025).

Optimal ternary cyclic codes have significant utility in practical domains due to algebraic structure—supporting fast encoding/decoding—and maximal error correction relative to size/rate. Applications include robust storage and communication systems, combinatorial designs, authentication codes, and cryptographic systems (Ding et al., 2013, Ding et al., 2013).

Recent progress continually extends the known taxonomy of such codes. Wu et al. introduced novel classes with three zeros and dimensional variants, demonstrating further nonequivalence and optimality (Wu et al., 2024). Additionally, new square-root-like bounds are derived for codes of dimension $\sim n/2$ in related work (Chen et al., 2023).

7. Open Problems and Future Directions

Several classification and parametrization problems remain unresolved:

• Complete algebraic characterization of all exponents $e$ (for fixed $m$) yielding optimal codes beyond known monomials, especially for multi-zero cases.
• Improved understanding of the solution profiles to critical equations $(x+1)^e \pm x^e \pm 1 = 0$ to pre-emptively rule in/out admissible exponents for arbitrary $(m,e)$ (Bao et al., 18 Jan 2026).
• Exploration of constructions over extensions involving more general functions, or leveraging new algebraic or combinatorial design methods.
• Investigation of deep connections with APN/planar functions, combinatorial structures, and implications for cryptographic resistance.

Addressing these will further enhance both the theoretical foundations and practical reach of optimal ternary cyclic codes.