[Paper Review] Universal Bounds for Size and Energy of Codes of Given Minimum and Maximum Distances

Peter Boyvalenkov, Peter D Dragnev+3 more|arXiv (Cornell University)|Oct 16, 2019

Coding theory and cryptography18 references5 citations

TL;DR

This paper establishes universal upper bounds on the size and lower bounds on the energy of codes with specified minimum and maximum distances in Hamming spaces using signed measures and positive definite polynomials. It derives explicit, computable bounds via Levenshtein-type quadrature formulas, proving optimality conditions and verifying tightness on known codes like ovoids and Golay code projections.

ABSTRACT

We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and cardinality. The distance distributions of codes that attain the bounds are found in terms of the parameters of Levenshtein-type quadrature formulas. Necessary and sufficient conditions for the optimality of our bounds are derived. Further, we obtain upper bounds on the energy of codes of fixed minimum and maximum distances and cardinality.

Motivation & Objective

• To derive universal upper bounds on the maximum size of codes with given minimum and maximum distances in Hamming spaces.
• To establish universal lower bounds on the potential energy of codes with fixed cardinality and maximum distance.
• To characterize the distance distributions of codes that attain these bounds using Levenshtein-type quadrature formulas.
• To provide necessary and sufficient conditions for the optimality of the derived bounds.
• To verify the tightness of the bounds on known optimal codes such as ovoids and Golay code projections.

Proposed method

• Employ signed measures that are positive definite up to certain degrees to construct linear programming bounds.
• Use Krawtchouk polynomial expansions to express polynomials in the dual space, with coefficients constrained to be non-negative.
• Define the feasible sets $\mathcal{F}_{n,\ell,s}$ and $\mathcal{G}^{(h)}_{n,\ell}$ for cardinality and energy bounds, respectively.
• Derive explicit formulas for the bounds using parameters $n$, $q$, $\ell$, and $s$, with $L = nq^2(1-s)(1-\ell)$.
• Apply Levenshtein-type quadrature formulas to characterize distance distributions of optimal codes.
• Verify optimality by checking that the distance distributions satisfy the system (38) and that the bounds are attained with equality.

Experimental results

Research questions

• RQ1 What are the tightest possible universal upper bounds on the size of codes with given minimum and maximum distances in $F_q^n$?
• RQ2 How can universal lower bounds on the potential energy of codes be derived for absolutely monotone interaction functions?
• RQ3 What are the necessary and sufficient conditions for a code to achieve the derived bounds?
• RQ4 Which known optimal codes (e.g., ovoids, Golay code projections) satisfy the optimality conditions and attain the bounds?
• RQ5 How can the distance distribution of codes achieving the bounds be explicitly computed?

Key findings

• The paper derives a universal upper bound for $\mathcal{A}_q(n,\ell,s)$ given by $\mathcal{A}_q(n,\ell,s) \leq \frac{L}{L + 4(q-1)(1-n) + 2nq(q-1)(s+\ell)}$, where $L = nq^2(1-s)(1-\ell)$, which generalizes Levenshtein bounds to codes with both minimum and maximum distance constraints.
• For codes with $\ell > -1$, the bound is tight and attained by known constructions such as the $[5,4,2]$ binary code, the $[6,3,2]$ binary code, and the Hill ternary and quaternary projective caps.
• The distance distribution of codes achieving the bound is fully characterized via the system (38), with $A_\ell = n(q-1)$ and $A_s = nd$ for ovoids in $\mathrm{PG}(3,q)$, yielding $|C| = q^4$.
• The $h$-energy of such optimal codes is computed as $E_h(C) = q^4(q^2+1)(q-1)\left(h\left(\frac{1-q^2}{1+q^2}\right) + h\left(\frac{1+2q-q^2}{1+q^2}\right)\right)$, and it attains the lower bound for all absolutely monotone $h$.
• The binary Golay code projections (lengths 23 and 22) achieve the bound $L_4(n,\ell,s)$ for $k=2$, with $|C| = 2^{11}$ and $2^{10}$ respectively, confirming the bound's tightness in higher-order cases.
• For the infinite family of codes from Dodunekov, Helleseth, and Zinoviev, the bound is attained when $N \geq 1 + (q^m - q)/2$, with $|C| = q^{2m} = L_2(n,\ell,s)$, and their distance distributions are explicitly computed via the system (38).