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Determining the covering radius of all generalized Zetterberg codes in odd characteristic

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Pith's one-line read This paper solves the open problem of determining the covering radius of every generalized Zetterberg code in odd characteristic, proving it is 2 or 3 with an explicit criterion.

desk verdict The Section 3 machinery is solid and the main open case is genuinely addressed, but the paper overclaims completion when Example 4.15 leaves two small covering radii undetermined, and the core Theorem 4.2 is not proved here. read the letter →

arxiv 2411.14087 v1 pith:CUVQ2BF2 submitted 2024-11-21 cs.IT math.IT

Minjia Shi , Shitao Li , Tor Helleseth , Ferruh Ozbudak This is my paper · ORCID
classification cs.ITmath.IT MSC 94B05
keywords coveringradiusgeneralizedZetterbergcodestwistedhalfquasi-perfectWeilsumsalgebraiccurvesoverfinitefieldsoddcharacteristic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

A code's covering radius is the largest distance any received word can be from the code and still be correctable; knowing it tells you how many errors a decoder must handle. This paper closes the last open case for the generalized Zetterberg codes over finite fields of odd characteristic, the case where the field size $q_0^s$ is congruent to $7$ mod $8$. It proves the covering radius is at most $3$, gives a criterion for when it is exactly $3$ rather than $2$, and shows that all sufficiently large odd extension degrees give radius $3$. It also introduces twisted half generalized Zetterberg codes and proves they inherit the same covering radius, yielding quasi-perfect codes when the radius is $2$. Together with the earlier work [31], this settles the covering radius for every generalized Zetterberg code in odd characteristic.

What carries the argument

The engine is a translation of the covering-radius question into solvability of systems of equations over finite fields, and then into rational points on a fiber-product curve. For the upper bound, the code has covering radius at most $3$ if every nonzero field element is a sum of three elements from the subgroup $H_m$ of $\mathbb{F}_{q^2}^*$ of order $m(q+1)$, where $m=(q_0-1)/2$; this is the family of “Property $P_i$” conditions indexed by the $2^\ell$-th roots of unity. Each such condition reduces to finding $x_1,x_2,x_3,y_1,y_2,y_3 \in \mathbb{F}_q$ with prescribed sums and each $x_j^2+y_j^2$ equal to $1$ (or to a fixed nonsquare $D$). Weil sums bound the number of admissible $x_1$'s, with the exceptional values $\alpha=\pm1$ treated separately. For exactness, “Property $\mathrm{NP}_i$” asserts that some $\gamma$ with $\gamma^q = \theta^i \gamma$ is not a sum of two elements of $H_m$; this is equivalent to the unsolvability of a system that, for even $i$, becomes the existence of $x,y_1,\ldots,y_m \in \mathbb{F}_q^*$ with $y_j^2 = x^2 - \alpha_j$ for all nonzero squares $\alpha_j$ in $\mathbb{F}_{q_0}$, with an analogous nonsquare version for odd $i$. The fiber-product curve $\chi: y_j^2 = x^2 - \alpha_j$, $j=1,\ldots,m$, has genus $1+2^{m-1}(m-2)$; the Hasse–Weil bound on its rational points yields the threshold $s^*$ beyond which solutions certainly exist, forcing radius $3$.

What would settle it

Check a currently undecided finite case, for instance $q_0=31$, $s=7$ or $q_0=47$, $s=7$: determine whether the system $y_i^2=x^2-\alpha_i$, $i=1,\ldots,m$, has a solution with $x,y_1,\ldots,y_m\in\mathbb{F}_{q_0^s}^*$, and compare with the covering radius of $\mathcal{C}_s(q_0)$ computed by exhaustive search. If the radius is not exactly 3 when the system is solvable, the paper's central dichotomy fails. For the large-$s$ claim, recompute the genus of the fiber-product curve directly; any value different from $1+2^{m-1}(m-2)$ invalidates the Hasse–Weil bound that produces $s^*$.

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Extended reading notes

Core claim

The paper establishes that when $q_0^s \equiv 7 \pmod 8$, equivalently $q_0 \equiv 2^\ell - 1 \pmod{2^{\ell+1}}$ with $s$ odd, the generalized Zetterberg code $\mathcal{C}_s(q_0)$ of length $q_0^s+1$ over $\mathbb{F}_{q_0}$ has covering radius at most $3$ (Corollary 3.14). It is exactly $3$ precisely when at least one of the properties $\mathrm{NP}_i$ holds, and exactly $2$ otherwise (Theorem 4.2). The case $s=1$ always gives radius $2$ (Corollary 4.7), while every odd $s$ at or above an explicit threshold $s^*$ gives radius $3$ (Theorem 4.8). The threshold comes from a Hasse–Weil lower bound on the number of $\mathbb{F}_{q_0^s}$-rational points of a fiber-product curve $y_i^2 = x^2 - \alpha_i$, where $\alpha_i$ runs through the nonzero squares of $\mathbb{F}_{q_0}$. The twisted half generalized Zetterberg codes of length $(q_0^s+1)/2$ have the same covering radius as the full codes, and because their packing radius is $1$, they are quasi-perfect exactly when the covering radius is $2$ (Theorem 5.2).

Load-bearing premise

The load-bearing premise is that the auxiliary curve $y_i^2 = x^2 - \alpha_i$ for $i=1,\ldots,m$ has exactly the genus $1+2^{m-1}(m-2)$ quoted from the earlier paper, and the entire threshold argument for large $s$ imports that number without recomputing it.

Editorial extensions

If this is right

  • For every finite field of odd characteristic and every $s\ge1$, the covering radius of $\mathcal{C}_s(q_0)$ is now an explicit value, 2 or 3, decided by whether the appropriate $\mathrm{NP}_i$ property holds.
  • When $s=1$, the twisted half code has parameters $[(q_0+1)/2,(q_0-3)/2,3\le d\le4]$ and covering radius 2, so it is quasi-perfect.
  • For each base field with $q_0\equiv2^\ell-1\pmod{2^{\ell+1}}$, all odd $s\ge s^*$ give covering radius 3, with $s^*$ defined explicitly by a single inequality.
  • If an odd extension degree $s$ gives radius 3, then every odd multiple $st$ also gives radius 3 (Proposition 4.11).
  • Twisted half generalized Zetterberg codes inherit the full code's covering radius, so the construction supplies quasi-perfect codes whenever the radius is 2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the unresolved small cases it lists, such as $q_0=31$, $s=7$ or $q_0=47$, $s=7,9$, can be decided not by exhaustive column searches but by checking whether the fiber-product curve has an $\mathbb{F}_{q_0^s}$-rational point above some $u\in\mathbb{F}_q$, i.e. whether $N(s)>0$.
  • If the quoted genus $1+2^{m-1}(m-2)$ is correct, the same Hasse–Weil counting gives explicit thresholds for every base field in the congruence class, and the examples $q_0=7,23,31,47$ are just the smallest instances of a uniform phenomenon.
  • The dichotomy between radius 2 and radius 3 suggests a sharper structural fact: the covering radius is decided by a single obstruction, namely whether one auxiliary element $\gamma$ fails to be a sum of two elements of $H_m$, so computing $\rho$ is equivalent to solving one system of equations rather than an optimization over all received words.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies generalized Zetterberg codes C_s(q0) over finite fields of odd characteristic in the remaining congruence class q0^s ≡ 7 (mod 8). The authors prove that the covering radius of such a code is at most 3, give an exact dichotomy (radius 3 if a certain Property NP_i holds and 2 otherwise), show that the radius is 2 for s = 1, and prove that for odd s beyond a threshold s* defined in Eq. (45) the radius is 3. They also define twisted half generalized Zetterberg codes and show that these codes inherit the same covering radii, yielding quasi-perfect codes for some parameters. The main techniques are elementary finite-field arithmetic, decomposition of F_{q^2} into cosets, Weil sums, and a Hasse-Weil estimate on a fiber product of curves.

Significance. If the arguments are completed and the overclaim is corrected, this is a substantial contribution: it would settle the open problem from [31] for the remaining congruence class, up to a small number of explicit undecided cases, and it would provide the first exact covering-radius results for this family in the hard case. The paper contains genuinely self-contained parts, especially the upper bound ρ ≤ 3 in Section 3, and it gives a concrete system-of-equations criterion for the exact radius. The introduction of twisted half generalized Zetterberg codes is a useful by-product and yields new quasi-perfect codes. However, the paper as written claims more than it proves: its own examples leave some covering radii undetermined, and a few load-bearing proof steps are skipped or quoted from [31].

major comments (4)
  1. [Abstract; Section 4, Examples 4.14–4.15] The paper's central claim that the covering radius of every generalized Zetterberg code in odd characteristic is now determined is not supported by the manuscript's own examples. Example 4.15 explicitly states that membership of 7 and 9 in I(47) is unknown, so the covering radii of C_7(47) and C_9(47) are not determined; Example 4.14 leaves the status of 7 in I(31) open. Since I(q0) is defined as the set of odd s with ρ(C_s(q0)) = 3, and Theorem 4.2 only gives a criterion that is not evaluated for these small s, the abstract's claim and the sentence in Section 4 that 'We solve this open problem completely' overstate the result. Please either settle these finitely many undecided cases or revise the claims throughout to say that the covering radius is determined for all codes except possibly an explicit finite list, or 'up to a finite computation'.
  2. [Theorem 4.2] Theorem 4.2 is the central characterization of when the covering radius equals 3, yet its proof is skipped with the remark that it uses the same arguments as [31]. Because [31] explicitly left the case q0^s ≡ 7 (mod 8) open, this is not literally a theorem proved in [31], and the step from the upper bound ρ ≤ 3 in Corollary 3.14 to the exact dichotomy 'ρ = 3 if and only if some Property NP_i holds' is not immediate. Please supply a proof or a detailed reduction, in particular showing how the results of Section 3 and the properties of H_m combine to rule out ρ ≤ 2 when no NP_i holds.
  3. [Theorem 3.12, Eq. (36)] The Weil bound for the sum of η((1 − x^2)Δ(x)) is asserted without proving that the degree-6 polynomial has no square factor; Proposition 3.8 only establishes that Δ(x) itself is not a square. If (1 − x^2)Δ(x) were a non-square constant times a square, the sum would be of size ≫ q rather than O(√q), and the positivity estimate N1 > 0 for q > 94 would be invalid. This is repairable: one can show Δ(±1) = 0 and that the remaining quadratic factor has discriminant 4(α^2 + 1)/α^2 ≠ 0, so the polynomial is not a constant times a square, but the argument should be included in the paper.
  4. [Theorem 4.8, Eq. (48)] The Hasse-Weil lower bound in Eq. (48) uses the genus of the fiber product χ, quoted as 1 + 2^{m−1}(m − 2) from 'the proof of [31, Theorem IV.1]'. This genus value is load-bearing for the estimate N(χ;s) > 2^m and hence for the threshold s* in Eq. (45), but it is not recomputed and no precise theorem statement is cited. Please either reproduce the genus computation or cite the exact numbered statement in [31] where this formula is proved, and confirm that it applies to the curve as defined here for all m.
minor comments (4)
  1. [Theorem 3.12] The list of small q for which a direct Magma check is used includes 87, but 87 is not a prime power and no field F_87 exists; the valid set is {7, 23, 31, 47, 71, 79}.
  2. [Theorems 3.12–3.13 and Examples 4.12–4.15] The Magma computations are described only in words, and some are said to be 'extremely time consuming' or to have been stopped. Please provide the Magma scripts or enough computational details to make the finite-field checks reproducible.
  3. [Throughout] There are several typos: 'Theoprem' for 'Theorem' in the citations in Theorem 3.12, 'D1 and D2 are are nonzero squares' in Proposition 4.3, 'Zettenberg' for 'Zetterberg' in Definition 5.1, and 'folows' for 'follows' in Theorem 5.2.
  4. [Example 4.14] The statement 'I(31) = {s ≥ 5 : s odd} or I(31) = {s ≥ 9 : s odd} ∪ {5}' is logically two alternatives; it would be clearer to write that I(31) is one of these two sets, since the 7 ∈ I(31) question is explicitly left open.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the upper-bound proof is self-contained, and reliance on [31] is prior-work scaffolding rather than a self-assumed conclusion; however the paper's own Example 4.15 leaves I(47) unresolved, so the unqualified 'all codes' claim is a scope/correctness gap, not circularity.

full rationale

The derivation chain is not circular. Section 3 proves the main upper bound rho(C_s(q0)) <= 3 directly: Propositions 3.3-3.10 reduce Property P_i to systems over F_q, Theorems 3.12 and 3.13 use Weil sums with the exceptional small fields checked by Magma, and Corollary 3.14 follows. No parameter is fitted and no prediction is a renamed input; the threshold s* in Eq. (45) is defined by an explicit inequality, not by data fitting. Theorem 4.2 is stated with proof deferred to 'the same arguments in [31]', but it is a characterization: given rho <= 3, rho = 3 exactly when some syndrome is not a sum of two parity-check columns, which is Definition 4.1 after the decomposition in Theorem 3.2; it does not assume the target open case from [31]. Theorem 4.8 uses the genus of the fiber product curve, quoted from [31, Theorem IV.1], and [31, Theorem VI.4] is used for the twisted-half codes; these are auxiliary geometric and prior-code results, not equivalents of the covering-radius conclusion, so the self-citations are not circular. The paper does, however, contain a stated limitation that conflicts with the abstract's 'all generalized Zetterberg codes' claim: in Example 4.15 the authors write 'we do not know if 7 is in I(47) or not. Similarly we do not know if 9 is in I(47) or not. Hence we determine that I(47) = {s >= 11 : s is an odd integer} union J, where J is a subset of {7,9}', and Example 4.14 likewise leaves 7 in I(31) undecided. Thus the exact covering radii of C_7(47), C_9(47), and possibly C_7(31) are not determined by the paper. This is a scope/correctness gap, not circularity, and it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two mathematical facts imported from prior work: the genus of the auxiliary curve (from [31]) and the validity of Weil-type character-sum bounds for the specific polynomials in Section 3. There are no free parameters fitted to data. No new physical or mathematical entities are postulated beyond the code definitions themselves, which are standard linear codes.

assumptions (4)
  • domain assumption The genus of the fiber product χ (defined by y_i^2 = x^2 - α_i for i=1..m) equals 1 + 2^{m-1}(m-2).
    Quoted from [31, Theorem IV.1] and used in Theorem 4.8 to apply the Hasse-Weil inequality; not re-derived in this paper.
  • domain assumption The character sums over F_q of η(Δ(x)) and η((1-x^2)Δ(x)) satisfy the Weil bounds 3√q and 5√q respectively.
    Used in Theorem 3.12; the paper proves Δ is not a square (Prop 3.8) but does not explicitly rule out a square factor for (1-x^2)Δ(x).
  • ad hoc to paper The Magma computations for the small fields q ∈ {7,23,31,47,71,79} correctly verify the required properties.
    Theorem 3.12 relies on a direct search for q < 94; no code or output is supplied.
  • standard math Hasse-Weil inequality for the fiber product curve χ over F_q.
    Standard theorem in algebraic geometry over finite fields, used in Theorem 4.8.

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Pith. "Pith review of Determining the covering radius of all generalized Zetterberg codes in odd characteristic." pith.science (2026). https://pith.science/paper/CUVQ2BF2

@misc{pith2026241114087,
  author       = {Pith},
  title        = {Pith review of: Determining the covering radius of all generalized Zetterberg codes in odd characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUVQ2BF2}},
  note         = {Machine review of arXiv:2411.14087}
}
abstract

For an integer $s\ge 1$, let $\mathcal{C}_s(q_0)$ be the generalized Zetterberg code of length $q_0^s+1$ over the finite field $\F_{q_0}$ of odd characteristic. Recently, Shi, Helleseth, and \"{O}zbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of $\mathcal{C}_s(q_0)$ for $q_0^s \not \equiv 7 \pmod{8}$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for $q_0^s \equiv 7 \pmod{8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length $\frac{q_0^s+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.

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Reference graph

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