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Further Study of p-Ary Self-Orthogonal Codes From Square Functions

Tingting Wu, Xianhong Xie

arXiv:2610.35346v1cs.ITmath.IT

TL;DR

The paper addresses whether a double-trace square-function construction can extend from ternary codes to arbitrary odd prime fields and how the defining parameter affects code properties. It analyzes quadratic trace codes to determine weight distributions, extends the treatment to both extension-degree parities, and studies associated punctured codes.

  • Problem

    A prior double-trace square-function construction treated ternary codes; the paper asks whether it extends to arbitrary odd prime fields and how the defining parameter affects code properties.

    Source: §1 · p. 2

  • Method

    The paper studies p-ary trace codes defined by two quadratic trace conditions, using quadratic character sums and quadratic trace-equation solution counts to determine weight distributions.

    Source: §1 · pp. 1–2

  • Results

    The treatment extends the construction to arbitrary odd primes and both odd and even extension degrees, while determining complete weight distributions.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The defining set's F*_p-orbit structure yields punctured codes with determined parameters, weight distributions, and dual distances.

    Source: §1 · p. 2

    The stated setting is finite fields of odd characteristic, with p an odd prime and q = p^m > 1.

    Source: §2.1 · p. 2

Abstract

from arXiv · show

In this paper, we investigate a family of trace codes over finite fields of odd characteristic defined by two quadratic trace conditions. By analyzing the quadratic forms associated with these trace conditions and evaluating the corresponding character sums, we obtain the parameters and complete weight distributions of the codes. We further exploit the natural multiplicative orbit structure of the defining sets to obtain punctured codes and determine their parameters, weight distributions and dual distances. Moreover, we establish self?orthogonality criteria and investigate divisibility and projectivity of the constructed codes. These results give a unified treatment for arbitrary odd prime fields and for extension degrees of both parities.

1 Introduction

The paper extends a double-trace square-function construction beyond the ternary setting and studies how field arithmetic shapes the resulting codes. It determines weight distributions and structural properties, including those of associated punctured codes.

1 Introduction

  • The double-trace square-function construction is extended from the ternary even-degree setting to arbitrary odd primes and both odd and even extension degrees.The paper investigates p-ary trace codes defined by two quadratic trace conditions.Source: §1 · p. 2
  • Quadratic character sums and solution counts for systems of quadratic trace equations are used to determine complete weight distributions.Source: §1 · p. 2
  • Code parameters and weight distributions depend on the intermediate field Fp(a) and quotient e = m/d, with separate cases by the parities of m and e.Source: §1 · p. 2
  • The F*_p-orbit structure of the defining set yields punctured codes whose parameters, weight distributions, and dual distances are determined.Source: §1 · p. 2
  • The paper establishes self-orthogonality criteria and examines divisibility and projectivity for the codes and their punctured versions.Source: §1 · p. 2

2 Preliminaries

The preliminaries set notation for finite fields and quadratic characters, introduce the projective line over Fp, and collect tools for evaluating quadratic sums.

2.1 Notations

  • The notation fixes an odd prime p, q = p^m, the absolute trace, and the quadratic and additive characters and Gauss sums used over the fields.The norm and matrix transpose are also specified.Source: §2.1 · pp. 2–3

2.2 Projective line

  • Points of P1(Fp) have representatives [t : 1] or the point at infinity, giving p + 1 points; PGL2(Fp) acts by Möbius transformations.Source: §2.2 · p. 3

2.3 Quadratic Gauss sums

  • The section defines quadratic Gauss sums and provides lemmas for their evaluation and for sums involving quadratic polynomials and quadratic forms.The quadratic-form lemma also identifies γ as a diagonal multiplier of Q.Source: §2.3 · pp. 3–4

3 The code CDa from double trace functions

This section analyzes the double-trace code using non-degenerate quadratic forms, treating even and odd extension degrees separately to derive code parameters and weight or distance results under stated conditions. For odd extension degree, it also gives a weight specialization based on p modulo 4 and identifies illustrative codes for 2 | e and d = 3.

3 The code CDa from double trace functions

  • For every nonzero c = u + va, the quadratic form x ↦ Tr(cx^2) is non-degenerate of rank m.The section also states that the discriminant of x ↦ Tr(x^2) is a square class for odd m and a nonsquare class for even m.Source: §3 · p. 4

3.1 The case of even m

  • For even m, Theorem 3.7 gives dim CDa = m − log_p H and characterizes the nonzero weights and exact minimum distance.Here H is the size of the kernel of the evaluation map.Source: §3.1 · p. 8
  • The reported code weight enumerator is WCDa(z) = 1 + 52z^54 + 312z^66 + 208z^72 + 156z^78.Source: §3.1 · p. 8
  • When m is even and at least 6, the evaluation map is injective; it is also injective for p ≥ 11, while m = 2 and m = 4 yield zero codes in the stated cases.For m = 2 and m = 4 with d = 2, the length and dimension are both zero.Source: §3.1 · pp. 9–10

3.2 The case for odd m

  • For odd m, the minimum distance is obtained by maximizing the reachable values of s_a(β); when d = m and e = 1, only nonzero square β-values are reachable.In that special case each such β has two preimages.Source: §3.2 · p. 13
  • For m = 3, CDa is a [p − 1, 1, p − 1]_p code with minimum distance p − 1 and nonzero-weight frequency p^2(p − 1).The zero-weight frequency is p^2.Source: §3.2 · p. 13
  • The scalar multiplication action is free and preserves both trace constraints, so the set forms a single orbit.The passage states that |D_a| = p − 1.Source: §3.2 · p. 13
  • When p ≡ 1 or 3 (mod 4), the kernel value is p or −p, yielding weights 0 and p − 1, respectively.Source: §3.2 · p. 13

3.3 Illustrative codes

  • The subsection presents illustrative codes for the cases 2 | e and d = 3.Source: §3.3 · p. 14

A. The case for even e

For even e, the section derives code parameters and minimum-distance results from trace conditions and their quadratic forms. It also gives cubic-extension character-sum distributions and examines orbit structure and punctured-code self-orthogonality.

A. The case for even e

  • Every nonzero Fp-linear combination of Ω has quadratic character 1 in Fq.Source: A. The case for even e · p. 15
  • For even e and m ≥ 6, Theorem 3.18 gives the code parameters and minimum distance of C_Da.The supplied passages do not include the formulas for these values.Source: A. The case for even e · p. 15
  • When δ = 1 and d = 2, the p + 1 elements of Ω represent all projective points, so every nonzero linear functional vanishes at one point and the minimum distance is w1.In this case M0 = 0 and M1 > 0.Source: A. The case for even e · p. 16
  • For cubic extensions, the distribution of s_a(β) depends only on p, not on the minimal polynomial or the listed orbits of a.The theorem gives the distribution in Table 1; the provided passages do not include its entries.Source: A. The case for even e · p. 20, Table 1 · p. 20
  • Among nonzero β ∈ K, p^2 − 1 quadratic forms Q_β have rank 1, while the remaining p^3 − p^2 have rank 2; rank-2 values split evenly between squares and nonsquares.The rank-1 forms contribute zero to the character sum.Source: A. The case for even e · pp. 21–22
  • The PGL2(Fp) action on K outside Fp is free and transitive, so all elements there lie in a single orbit.The stabilizer of each such element is trivial, and the orbit size is p^3 − p.Source: A. The case for even e · p. 22
  • For m = 3e with odd e ≥ 3, C_Da has parameters [p^(m−2) − 1, m]_p, with its weight enumerator and minimum distance determined.The supplied passages give the parameter statement but not the weight-enumerator or distance formulas.Source: A. The case for even e · pp. 22–23

4 More properties for CDa

The section studies further properties of CDa and its punctured codes, including self-orthogonality criteria, dual distances, and divisibility. It also presents weighted-code criteria and examples for these properties.

4 More properties for CDa

  • The punctured code CRa has length #Da/(p−1) and dimension equal to dim(CDa), namely m−logp H.Theorem 4.1 also states that its weight frequencies satisfy a formula not included in the supplied passage.Source: §4 · p. 23

4.1 Self-orthogonality

  • CDa is self-orthogonal exactly when p > 3, or when p = 3 and m ⩾7 for odd m or m ⩾6 for even m.CRa is self-orthogonal under exactly the same conditions as CDa.Source: §4.1 · pp. 23, 26
  • A monomially equivalent weighted punctured code is self-orthogonal if and only if nonzero-square weights can be chosen to satisfy the stated Gram-matrix condition.The property is independent of the basis and initial projective representatives; the supplied passages do not include the condition’s full formula.Source: §4.1 · p. 26
  • For p = 3 and a ∉ F3, solvability requires m ≥ 7 when m is odd and m ≥ 6 when m is even; for p > 3, m = 2 gives an empty defining set and m = 3 yields a [1, 1, 1]p punctured code that is never solvable.These are the stated low-dimensional cases in the self-orthogonality classification.Source: §4.1 · p. 30
  • The adjoint of La is an evaluation map; when a ∉ Fp, Q0 and Qa are independent quadratic forms in I2(Xa), while Xa = ∅ implies rank(La) = 0.The stated conclusions do not depend on the basis of Fq/Fp or the initial projective representatives.Source: §4.1 · p. 31
  • For p = 5, 7, and p ≥ 11, the norm construction yields self-orthogonal weighted punctured codes when m ≥ 11, m ≥ 8, and m ≥ 7, respectively.The construction is independent of d = [Fp(a) : Fp] and remains effective when geometric degeneration occurs.Source: §4.1 · p. 33

4.2 Dual distance

  • In CDa, distinct coordinates from the same Fp* orbit yield weight-2 dual codewords; in CRa, the generator columns represent distinct projective points and no two are linearly dependent.Three-weight dual codewords correspond to projective lines in Xa, and their count is obtained by summing over those lines.Source: §4.2 · p. 34
  • The punctured dual-distance classification treats m ≥ 7 for odd p and a ∉ Fp, m = 5 with d = 5, m = 6 with d ∈ {2, 3, 6}, and m = 4.For m = 6, the supplied passage states that all three layers satisfy the theorem's conclusion, but does not provide the values.Source: §4.2 · pp. 36–37
  • For d = 2, Xa = Xa(Fp) = ∅, and the punctured code is the zero code.Source: §4.2 · p. 37
  • For the smooth three-dimensional complete intersection of two quadrics Xa over Fp, a Fp-rational point of its Hilbert scheme of lines yields a line defined over Fp and therefore Λa > 0.Source: §4.2 · p. 38
  • When s = #Xa(Fp) ≥ 5, the dual punctured code has minimum distance 4; the same holds for nonzero dual codes with s ≤ 4.If s = dim(CRa), the dual code is zero.Source: §4.2 · p. 40

4.3 Divisibility

  • For a ∉ Fp, the code is p-divisible when m ≥ 7 is odd or m ≥ 6 is even.The section also reports examples with ΔR = 1 and ΔD = p − 1 for m = 3, and with greatest common divisor of weights equal to 1 for p = 3, 5, m = d = 4 and p = 3, m = 6, d = 2, 3, 6.Source: §4.3 · pp. 40–41
  • The construction’s self-orthogonality results show that containing the all-ones vector together with p-divisibility, though sufficient in general, is not necessary here.The cited general criterion is [12, Theorem 11].Source: §4.3 · p. 41
  • For odd m, gcd(P − Bm, P, P + Bm) = Bm = p^((m−3)/2), with the middle weight associated with the square locus retaining an extra factor p.The section also establishes κo(a) = 0 in the full-degree case.Source: §4.3 · pp. 43–44
  • 1 + 3024z^702 + 13850z^729 + 2808z^756 is the complete weight enumerator WCRa(z) for p = 3, m = 9, d = e = 3.The example also reports a = ΔR 3^(m−5)/2 = 3.Source: §4.3 · p. 44
  • For even m with d = 2, e = 2 gives the empty-code case, while e ≥ 4 yields an extra gcd factor p; for odd e, κe(a) = 0 and (4.30) follows.The even-m theorem covers a ∉ Fp and e ≥ 2; its d = 2 cases are specified separately.Source: §4.3 · p. 45
  • For p = 3, m = 6, d = 3, e = 2, WCDa(z) = 1 + 52z^54 + 312z^66 + 208z^72 + 156z^78 and WCRa(z) = 1 + 52z^27 + 312z^33 + 208z^36 + 156z^39.These enumerators are reported from an exhaustive computation.Source: §4.3 · p. 45

5 Conclusions

The paper constructs p-ary trace codes from two quadratic trace conditions and analyzes their parameters and complete weight distributions. It extends the square-function double-trace construction to arbitrary odd primes and both parities of the extension degree, and studies orbit-induced punctured codes and their structural properties.

5 Conclusions

  • Quadratic forms and character sums yield parameters and complete weight distributions for codes defined by two quadratic trace conditions.The results describe the roles of d = [Fp(a) : Fp], e = m/d, and their parities.Source: §5 · p. 45
  • The multiplicative orbit structure yields punctured codes whose dual distances, self-orthogonality, divisibility, and projectivity are characterized.For p > 3, weighted self-orthogonality is reduced to ker La ∩ SXa p = ∅, with explicit existence ranges.Source: §5 · pp. 45–46
  • Unlike Xiang and Tang [20], which treats the double-trace square-function construction only for p = 3 and even m, this work covers arbitrary odd primes and both parities of m.The paper uses two quadratic trace equations as its construction framework, rather than bent functions, special defining sets, or p-divisibility as the main framework [7, 12, 19].Source: §5 · pp. 45–46
  • Further study may consider additional quadratic trace constraints and related quantum or LCD codes.Source: §5 · p. 46
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