pith. sign in

arxiv: 2604.21808 · v1 · submitted 2026-04-23 · 💻 cs.IT · math.IT

Recursive Structure of Hulls of PRM Codes

Yufeng Song , Qin Yue This is my paper

Pith reviewed 2026-05-08 13:47 UTC · model grok-4.3

classification 💻 cs.IT math.IT MSC 94B0511T71
keywords projective Reed-Muller codeshull dimensioncombinatorial numbersReed-Muller codesself-orthogonal codesfinite geometry codescoding theory
0
0 comments X

The pith

The hull dimension of a projective Reed-Muller code PRM(q,m,v) equals its code dimension minus a sum of combinatorial counts A_s(w) when v lies in the lower-half parameter range.

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

The paper determines an explicit formula for the dimension of the hull of the projective Reed-Muller code PRM(q,m,v). For v in the interval between r(q-1)/2 and (r+1)(q-1)/2, the hull dimension is the code dimension reduced by a sum over i of the terms A_{2i+epsilon}(v minus a multiple of (q-1)), where the A_r(v) are defined by an alternating double binomial sum. This covers the open range 0 < v < m q^m / 2, with the complementary upper range recovered from duality. A reader cares because the hull controls how much the code is self-orthogonal, which directly affects constructions that rely on these codes for secret sharing or stabilizer quantum codes.

Core claim

For nonnegative integer r and positive integer v satisfying r(q-1)/2 < v < (r+1)(q-1)/2, the dimension of the hull of PRM(q,m,v) equals dim PRM(q,m,v) minus the sum from i=0 to floor(r/2) of A_{2i+epsilon}(v - (floor(r/2)-i)(q-1)), where epsilon is 0 if r even and 1 if r odd, and the A_r(v) are the alternating binomial sums given in the paper; the identity holds for m at least r+1.

What carries the argument

The combinatorial numbers A_r(v), defined as 1 for r=0 and as a double alternating binomial sum for r>0, which serve as the correction terms that subtract the intersection dimensions from the code dimension.

If this is right

  • The hull dimension can now be evaluated by arithmetic on binomial coefficients rather than by enumerating the code.
  • The same formula immediately yields when the hull is trivial or when the code is self-orthogonal.
  • Combined with the known duality relation, the result gives the hull dimension for every v between 0 and m q^m.
  • The recursive structure isolates the contribution of each parity class of r, allowing incremental computation when r increases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same counting technique may extend to the hulls of other evaluation codes on projective varieties once analogous intersection numbers are identified.
  • Closed-form simplifications of the sum over A terms could produce asymptotic statements on hull size as m or q grows.
  • Verification on small parameters would also confirm whether the combinatorial identity A_r(v) continues to hold when m is only slightly larger than r.

Load-bearing premise

The numbers A_r(v) correctly count the dimensions of the intersections that determine the hull inside the stated range of v.

What would settle it

Direct computation of the hull dimension over a small finite field, for example q=3, m=3, r=1 and v=2, by linear-algebra rank on the generator matrix, then comparison against the numerical value predicted by the formula.

read the original abstract

For a nonnegative integer $r$ and a positive integer $v$ satisfying \[ \frac{r(q-1)}{2}<v<\frac{(r+1)(q-1)}{2}, \] we define the combinatorial numbers \[ A_r(v)= \begin{cases} \displaystyle \sum_{t=r(q-1)-v}^{v}\ \sum_{j=0}^{r}(-1)^j\binom{r}{j}\binom{t-jq+r-1}{r-1}, & r>0,\\[1.2ex] 1, & r=0. \end{cases} \] For the projective Reed-Muller code $\PRM(q,m,v)$, we determine its hull dimension: \[ \dim \Hull\bigl(\PRM(q,m,v)\bigr) = \dim \PRM(q,m,v) - \sum_{i=0}^{\ell}A_{2i+\epsilon}\bigl(v-(\ell-i)(q-1)\bigr), \] where \[ \ell=\Bigl\lfloor\frac r2\Bigr\rfloor,\qquad \epsilon= \begin{cases} 0, & r\ \text{is even}, 1, & r\ \text{is odd}. \end{cases} \] This formula applies in the open lower-half range $ 0<v<\frac{m\Qm}{2}, $ equivalently for $v\in I_r$ with $m\ge r+1$; the range $ \frac{m\Qm}{2}<v<m\Qm $ is then obtained by S\o rensen's duality theorem \cite{Sorensen}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims an explicit formula for the hull dimension of the projective Reed-Muller code PRM(q,m,v) in the regime r(q-1)/2 < v < (r+1)(q-1)/2 (with m ≥ r+1), given by dim PRM(q,m,v) minus a sum of the combinatorial quantities A_{2i+ε}(v - (ℓ-i)(q-1)) where ℓ = floor(r/2) and ε encodes parity; the quantities A_r(v) are defined by a double sum of alternating binomial coefficients (or 1 when r=0). The complementary range follows from Sørensen duality.

Significance. If the binomial-sum identity for A_r(v) holds, the result supplies a closed-form, parameter-free expression for hull dimensions of PRM codes over the stated range. This is a concrete advance in the structural theory of these codes, directly usable for determining self-orthogonality properties and for constructions that rely on hull dimension.

minor comments (3)
  1. [Abstract / Definition of A_r(v)] In the displayed formula for A_r(v), the lower summation limit r(q-1)-v must be shown to be a nonnegative integer under the hypothesis r(q-1)/2 < v; the manuscript should add a short sentence confirming this for the open interval.
  2. [Abstract] The notation mQm (appearing as the upper bound m q^m / 2) should be written explicitly as m q^m throughout the text and abstract for readability.
  3. [Main theorem statement] The statement that the formula applies for v in I_r with m ≥ r+1 is given; a brief remark on why m ≥ r+1 is required to keep all binomial arguments nonnegative would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful summary of our results and the recommendation of minor revision. The report correctly identifies the regime and the role of the combinatorial quantities A_r(v).

read point-by-point responses

  1. Referee: If the binomial-sum identity for A_r(v) holds, the result supplies a closed-form, parameter-free expression for hull dimensions of PRM codes over the stated range.

    Authors: The manuscript derives the stated double-sum expression for A_r(v) (r>0) directly from the recursive decomposition of the hull; the identity is not assumed but proven by induction on r together with the explicit basis for the projective Reed-Muller code. Consequently the hull-dimension formula is closed-form and parameter-free in the indicated range. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper explicitly defines the combinatorial numbers A_r(v) via a closed-form double sum of binomial coefficients with alternating signs (or the base case 1 for r=0). The claimed hull-dimension formula is then written directly as dim(PRM) minus a finite sum of these A quantities evaluated at shifted arguments. No parameter is fitted to data and then re-used as a prediction; the definition of A_r(v) is independent of the target dimension formula. Sørensen duality is invoked only to obtain the complementary range and is an external reference, not a self-citation chain. The derivation therefore remains self-contained against the stated combinatorial counts and does not reduce to a tautology or to any input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard properties of vector spaces of homogeneous polynomials, the definition of projective Reed-Muller codes, and the combinatorial identity that the alternating binomial sum A_r(v) equals the desired intersection dimension. No free parameters are introduced; the only background results are classical facts about binomial coefficients and the known duality theorem of Sørensen.

axioms (2)
  • standard math Binomial coefficient identities and inclusion-exclusion hold over the integers for the given range of indices.
    Invoked in the definition of A_r(v) for r > 0.
  • domain assumption Sørensen duality theorem relates the hull dimension in the upper half-range to the lower half-range.
    Used to extend the formula from 0 < v < m q^m / 2 to the complementary interval.

pith-pipeline@v0.9.0 · 5641 in / 1563 out tokens · 36906 ms · 2026-05-08T13:47:37.320089+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    NewgeneralizationsoftheReed-Mullercodes—PartI:Primitive codes,

    T.Kasami,S.Lin,andW.W.Peterson,“NewgeneralizationsoftheReed-Mullercodes—PartI:Primitive codes,”IEEE Trans. Inf. Theory, vol. 14, no. 2, pp. 189–199, 1968

    work page 1968

  2. [2]

    NewgeneralizationsoftheReed-Mullercodes—PartII:Nonprimitivecodes,

    E.Weldon,“NewgeneralizationsoftheReed-Mullercodes—PartII:Nonprimitivecodes,”IEEETrans. Inf. Theory, vol. 14, no. 2, pp. 199–205, 1968. RECURSIVE STRUCTURE OF HULLS OF PRM CODES 25

    work page 1968

  3. [3]

    Polynomial codes,

    T. Kasami, S. Lin, and W. W. Peterson, “Polynomial codes,”IEEE Trans. Inf. Theory, vol. 14, no. 6, pp. 807–814, 1968

    work page 1968

  4. [4]

    On generalized Reed-Muller codes and their relatives,

    P. Delsarte, J. M. Goethals, and F. J. MacWilliams, “On generalized Reed-Muller codes and their relatives,”Inf. Control, vol. 16, no. 5, pp. 403–442, 1970

    work page 1970

  5. [5]

    Linear codes and modular curves,

    Y. I. Manin and S. G. Vladut, “Linear codes and modular curves,”J. Sov. Math., vol. 30, no. 6, pp. 2611–2643, 1985

    work page 1985

  6. [6]

    Projective Reed-Muller codes,

    G. Lachaud, “Projective Reed-Muller codes,” inCoding Theory and Applications, Lecture Notes in Computer Science, vol. 311, pp. 125–129, 1988

    work page 1988

  7. [7]

    TheparametersofprojectiveReed-Mullercodes,

    G.Lachaud,“TheparametersofprojectiveReed-Mullercodes,”DiscreteMath.,vol.81,no.2,pp.217– 221, 1990

    work page 1990

  8. [8]

    ProjectiveReed-Mullercodes,

    A.B.Sorensen,“ProjectiveReed-Mullercodes,”IEEETrans.Inf.Theory,vol.37,no.6,pp.1567–1576, 1991

    work page 1991

  9. [9]

    Automorphism groups of homogeneous and projective Reed-Muller codes,

    T. P. Berger, “Automorphism groups of homogeneous and projective Reed-Muller codes,”IEEE Trans. Inf. Theory, vol. 48, no. 5, pp. 1035–1045, 2002

    work page 2002

  10. [10]

    WeightenumeratorsofReed-Mullercodesfromcubiccurvesandtheirduals,

    N.Kaplan,“WeightenumeratorsofReed-Mullercodesfromcubiccurvesandtheirduals,”inArithmetic, Geometry, Cryptography, and Coding Theory, Contemp. Math., vol. 722, pp. 59–78, 2017

    work page 2017

  11. [11]

    Maximum number of common zeros of homogeneous polynomials over finite fields,

    P. Beelen, M. Datta, and S. R. Ghorpade, “Maximum number of common zeros of homogeneous polynomials over finite fields,”Proc. Amer. Math. Soc., vol. 146, no. 4, pp. 1451–1468, 2018

    work page 2018

  12. [12]

    A combinatorial approach to the number of solutions of systems of homogeneous polynomial equations over finite fields,

    P. Beelen, M. Datta, and S. R. Ghorpade, “A combinatorial approach to the number of solutions of systems of homogeneous polynomial equations over finite fields,”Mosc. Math. J., vol. 22, no. 4, pp. 565–593, 2022

    work page 2022

  13. [13]

    On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes,

    S. R. Ghorpade and R. Ludhani, “On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes,”Adv. Math. Commun., vol. 18, no. 2, pp. 360–382, 2024

    work page 2024

  14. [14]

    Subfield subcodes of projective Reed-Muller codes,

    P. Gimenez, D. Ruano, and R. San-José, “Subfield subcodes of projective Reed-Muller codes,”Finite Fields Appl., vol. 94, Art. 102353, 2024

    work page 2024

  15. [15]

    A recursive construction for projective Reed-Muller codes,

    R. San-José, “A recursive construction for projective Reed-Muller codes,”IEEE Trans. Inf. Theory, vol. 70, no. 12, pp. 8511–8523, 2024

    work page 2024

  16. [16]

    Recursive decoding of projective Reed-Muller codes,

    R. San-José, “Recursive decoding of projective Reed-Muller codes,”IEEE Trans. Inf. Theory, early access, 2026, doi: 10.1109/TIT.2026.3667436

    work page doi:10.1109/tit.2026.3667436 2026

  17. [17]

    Structure of weighted projective Reed-Muller codes,

    J. Nardi and R. San-José, “Structure of weighted projective Reed-Muller codes,” arXiv:2603.24397 [cs.IT], 2026

    work page arXiv 2026

  18. [18]

    Constructions of good entanglement-assisted quantum error correcting codes,

    K. Guenda, S. Jitman, and T. A. Gulliver, “Constructions of good entanglement-assisted quantum error correcting codes,”Des. Codes Cryptogr., vol. 86, no. 1, pp. 121–136, 2018

    work page 2018

  19. [19]

    Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to quantum codes,

    Y. Gao, Q. Yue, X. Huang, and J. Zhang, “Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to quantum codes,”IEEE Trans. Inf. Theory, vol. 67, no. 10, pp. 6619–6626, 2021

    work page 2021

  20. [20]

    Hulls of Reed-Solomon codes via algebraic geometry codes,

    B. Chen, S. Ling, and H. Liu, “Hulls of Reed-Solomon codes via algebraic geometry codes,”IEEE Trans. Inf. Theory, vol. 69, no. 2, pp. 1005–1014, 2023

    work page 2023

  21. [21]

    Hulls of projective Reed-Muller codes over the projective plane,

    D. Ruano and R. San-José, “Hulls of projective Reed-Muller codes over the projective plane,”SIAM J. Appl. Algebra Geom., vol. 8, no. 4, pp. 846–876, 2024

    work page 2024

  22. [22]

    Quantum error-correcting codes from projective Reed-Muller codes and their hull variation problem,

    D. Ruano and R. San-José, “Quantum error-correcting codes from projective Reed-Muller codes and their hull variation problem,”J. Algebra Appl., vol. 24, nos. 13–14, Art. 2541009, 2025

    work page 2025

  23. [23]

    HullsofprojectiveReed-Mullercodes,

    N.KaplanandJ.-L.Kim,“HullsofprojectiveReed-Mullercodes,”Des.CodesCryptogr.,vol.93,no.3, pp. 683–699, 2025, doi: 10.1007/s10623-024-01543-2

    work page doi:10.1007/s10623-024-01543-2 2025

  24. [24]

    Polynomeshomogenesquis’annulentsurl’espaceprojectif𝑃 𝑚(F𝑞),

    D.-J.MercierandR.Rolland,“Polynomeshomogenesquis’annulentsurl’espaceprojectif𝑃 𝑚(F𝑞),”J. Pure Appl. Algebra, vol. 124, no. 1–3, pp. 227–240, 1998. 26 Y. SONG AND Q. YUE

    work page 1998

  25. [25]

    Hull parameters of projective Reed-Muller codes,

    Y. Song and J. Luo, “Hull parameters of projective Reed-Muller codes,”Des. Codes Cryptogr., vol. 94, Art. 57, 2026, doi: 10.1007/s10623-025-01789-4

    work page doi:10.1007/s10623-025-01789-4 2026