Length-Maximal Codes with Given Singleton Defect: Structure and Bounds

Abstract: We study the maximum length of $q$-ary codes as a function of alphabet size, code size, and Singleton defect. For an $(n, M, d)_q$ code with dimension $κ= \log_q M \ge 2$ and Singleton defect $s = n - \lceilκ\rceil + 1 - d$, we establish a \emph{maximal-arc-type bound}. For $M = q<sup>k$, we call codes with $n = (s+1)(q+1) + k - 2$ \emph{length-maximal}, and show such codes are necessarily symbol-uniform, have pairwise distances confined to ${d} \cup {n-k+3, \ldots, n}$, and satisfy the divisibility condition $(s+2) \mid q(q+1)$. An equivalent form yields an improved Singleton-type inequality extending a result of Guerrini, Meneghetti, and Sala for binary systematic codes. When $s \ge 2q$, the bound tightens to $n \le s(q+1)+k-1$; more finely, when $αq \le s &lt; (α+1)q$ for integer $α\ge 2$, it tightens to $n \le (s+2-α)(q+1)+α+k-3$, improving on the main bound by $(α-1)q$. We identify several conditions under which nonlinear codes satisfy the Griesmer bound, including: $d \le q<sup>2$; $s \le q-1$; $s \ge βq$ with $d \le βq<sup>2$; and a parametric family of binary conditions. We also show that near-length-maximal $A<sup>1$MDS codes of length $k+2q-1$ cannot exist for $k \ge 5$ when $q=2$, nor for $k \ge 7$ when $q=3$. For codes of non-integer dimension $κ\in (k, k+1)$, an analogous bound holds but is never attained. This forces the corresponding Singleton-type inequality one unit tighter than the integer-dimension case. For rational non-integer $κ$, our bounds specialise to a length bound for additive codes of fractional dimension, complementing recent geometric results on additive codes. Throughout, the results parallel the theory of maximal arcs. Whether length-maximal nonlinear codes can exist for parameter ranges within which no linear length-maximal codes exist is the principal open problem raised by this work.

• The paper establishes new tight combinatorial upper bounds on q-ary code length based on parameters like dimension, alphabet size, and Singleton defect.
• It shows that length-maximal codes are symbol-uniform with constrained Hamming distance spectra, drawing parallels with maximal arcs in finite geometry.
• The work extends classic Singleton and Griesmer bounds to both nonlinear and non-integer dimension codes, offering a unified structural analysis.

Length-Maximal Codes with Given Singleton Defect: Structure and Bounds

Introduction and Context

This work provides a comprehensive structural and combinatorial analysis of $q$-ary codes—both linear and nonlinear—with prescribed code size, alphabet, and Singleton defect. It delivers new tight upper bounds on code length as a function of these parameters, elucidates the internal structure of optimal codes, analyzes various defect regimes, and unifies the treatment of integer and non-integer code dimensions. The analysis demonstrates that the ubiquitous linear geometry-based maximal arc bound extends, with highly rigid structure, to general codes, and that the existence criteria, weight distributions, and sharpness of classical bounds (such as Singleton and Griesmer) are inherited in large measure by nonlinear codes under specific circumstances.

Main Results and Theoretical Advances

Maximal-Arc-Type Bound

The paper proves that for an $(n, q^k, d)_q$ code with dimension $\kappa = \log_q M \geq 2$ and Singleton defect $s = n - \lceil \kappa \rceil + 1 - d$, the following upper bound holds for all nonlinear codes:

$n \leq (s + 1)(q + 1) + k - 2.$

This generalizes the classical maximal arc bound from projective geometry—which characterizes the maximum lengths of linear codes—without any recourse to linearity or geometric constructions. The proof is entirely combinatorial, based on a sophisticated use of double-counting agreements and a structural analysis of coordinate symbol frequencies.

Codes attaining equality in this bound, termed length-maximal, are shown to be symbol-uniform (every symbol appears equally often in every coordinate) and possess tightly restricted distance spectra: every pairwise distance is either the minimum distance or one of the top $k-2$ possible values. Length-maximality further induces the divisibility constraint $(s + 2) \mid q(q + 1)$ and $s \leq q - 1$, paralleling the structural conditions for maximal arcs in the linear case.

Improved Singleton-Type Bound and Defect Stratification

Expressing the maximal-arc bound in terms of $d$ yields an improved Singleton-type inequality:

$$\left\lceil d \frac{q + 1}{q} \right\rceil \leq n - k + 2,$$

which significantly sharpens the classical Singleton bound, recovers recent results for systematic binary codes as a special case [Guerrini, Meneghetti, and Sala], and becomes tight across several parameter regimes.

For codes of large defect ($(n, q^k, d)_q$0), a stratified family of bounds is developed. When $(n, q^k, d)_q$1, the code length is bounded above by:

$(n, q^k, d)_q$2

and, for $(n, q^k, d)_q$3, by

$(n, q^k, d)_q$4

offering further refinements as $(n, q^k, d)_q$5 increases. The improvement over the main bound approximately scales with $(n, q^k, d)_q$6. The stratified bounds precisely reflect the combinatorial constraints imposed by large defect and minimal code overlap.

Structural Characterizations

A critical consequence of these results is the rigidity forced upon length-maximal codes. Not only are they symbol-uniform, but their spectrum of Hamming distances is strictly confined to $(n, q^k, d)_q$7, excluding most possible intermediate distances. This also restricts the possible set of weight distributions, aligning with highly symmetric combinatorial objects akin to maximal arcs.

Furthermore, the required divisibility condition $(n, q^k, d)_q$8 is weaker than, but reminiscent of, the linear maximal arc condition $(n, q^k, d)_q$9. Whether this is the sharp necessary condition for nonlinear codes remains open.

Codes of Non-Integer Dimension

When the code dimension $\kappa = \log_q M \geq 2$0 is non-integer, the analogous upper bound is never attained; specifically, for $\kappa = \log_q M \geq 2$1 the best possible length is

$\kappa = \log_q M \geq 2$2

always strictly below the integer-dimension ceiling by one. This manifests in a Singleton-type inequality one unit tighter than the integer-dimension case and implies the nonexistence of length-maximal codes with non-integer dimension—reinforcing the role of integer-dimension combinatorics in the structure of extremal codes. For rational $\kappa = \log_q M \geq 2$3, new explicit bounds are derived, complementing and in some cases improving known geometric bounds for additive (fractional-dimension) codes.

Griesmer Bound for Nonlinear Codes

Notably, the analysis identifies several parameter regimes—previously only established for systematic or linear codes—where nonlinear codes necessarily satisfy the Griesmer bound. These include:

• $\kappa = \log_q M \geq 2$4 (for $\kappa = \log_q M \geq 2$5)
• $\kappa = \log_q M \geq 2$6
• Large defect $\kappa = \log_q M \geq 2$7 and $\kappa = \log_q M \geq 2$8 (for relevant $\kappa = \log_q M \geq 2$9)
• A complete characterization for all binary codes with $s = n - \lceil \kappa \rceil + 1 - d$0
• Several parametric families for general and additive (fractional) codes

These regimes are systematically summarized and shown to be sharp, providing a unified extension of the Griesmer property through the nonlinear and non-integer landscape.

Nonexistence and Tightness for Near-Maximal Codes

The work gives nonexistence results for near-length-maximal binary and ternary codes (specifically, $s = n - \lceil \kappa \rceil + 1 - d$1MDS codes of length $s = n - \lceil \kappa \rceil + 1 - d$2), using the Hamming bound to show that these codes do not exist beyond modest dimensions ($s = n - \lceil \kappa \rceil + 1 - d$3 for $s = n - \lceil \kappa \rceil + 1 - d$4, $s = n - \lceil \kappa \rceil + 1 - d$5 for $s = n - \lceil \kappa \rceil + 1 - d$6), with the result generalizing to $s = n - \lceil \kappa \rceil + 1 - d$7-ary codes via Hamming ball volume computations.

Implications and Open Problems

Practical and Theoretical Impacts

These combinatorial bounds decisively constrain the possible parameters for (potentially nonlinear) codes with prescribed Singleton defect, guiding both constructions and nonexistence proofs. They also indicate that, despite the far greater freedom in the nonlinear setting as compared to the linear, the possible maximal-length codes remain subject to deep combinatorial constraints that mirror and extend those of finite projective geometry.

From a practical standpoint, these results provide a roadmap for code designers: attaining the length-maximal bound (or approaching it) necessarily forces symbol-uniformity and severe restrictions on distance spectra; in contrast, pushing code lengths for large-defect or non-integer-dimension codes is structurally obstructed. Furthermore, the established regimes where the Griesmer bound persists for nonlinear codes inform the selection of parameters in robust communication systems over finite alphabets.

Future Directions

Several open questions are highlighted:

• Odd $s = n - \lceil \kappa \rceil + 1 - d$8 nonexistence: For linear codes, maximal arcs are known not to exist for odd $s = n - \lceil \kappa \rceil + 1 - d$9 and $n \leq (s + 1)(q + 1) + k - 2.$0. Whether any genuinely nonlinear length-maximal codes exist in this range is open. A positive (or negative) resolution will significantly affect the linear/nonlinear comparison in extremal code theory.
• Unique extendability: The extension property for near-maximal nonlinear codes, which parallels a result of Barlotti for linear maximal arcs, remains mysterious and ripe for deeper combinatorial analysis.
• Divisibility criterion sharpness: Determining if $n \leq (s + 1)(q + 1) + k - 2.$1 is the strongest necessary condition for all length-maximal codes, or whether the more restrictive linear condition $n \leq (s + 1)(q + 1) + k - 2.$2 holds in general, is unresolved.
• AMDS codes and near-maximal cases: For higher defect and larger $n \leq (s + 1)(q + 1) + k - 2.$3, especially when the Hamming bound ceases to provide nonexistence, the maximal attainable lengths remain unknown.

The structural results (symbol-uniformity, defect preservation, and explicit bounds) established here form a substantial toolkit for attacking these and related problems. They point to a possibly narrow parameter window where nonlinear codes might diverge from linear and almost-linear analogs, and indicate that even in the nonlinear case, the combinatorial echoes of finite geometry are profound and omnipresent.

Conclusion

This paper delivers a rigorous extension of maximal-arc structure and bounds to nonlinear $n \leq (s + 1)(q + 1) + k - 2.$4-ary codes, identifies the precise combinatorial properties forced by length maximality, and rigorously classifies parameter regimes for which classical bounds such as Griesmer carry over. The work not only generalizes prior results and closes open cases (including for systematic codes and additive fractional-dimension codes), but also lays out a structured agenda for further advances in extremal coding theory, particularly with respect to the nonlinear and non-projective cases.

Reference:

"Length-Maximal Codes with Given Singleton Defect: Structure and Bounds" (2604.03784)