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arxiv: 2606.25109 · v1 · pith:24QCEUXXnew · submitted 2026-06-23 · 🧮 math.NT · math.CO

Binomial sequences over prime fields

Miguel Beltr\'a , Sara D. Cardell , Ver\'onica Requena This is my paper

Pith reviewed 2026-06-25 22:17 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords binomial sequencesp-ary sequencesprime fieldsvector space basisperiodic sequencesPascal's trianglelinear complexity
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The pith

Binomial p-ary sequences form a basis for the vector space of all sequences over F_p with period a power of p.

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

The paper extends the binary binomial sequences, defined as diagonals of Pascal's triangle modulo 2, to the setting of sequences over a prime field F_p. It examines the intrinsic properties and formation rules that define these p-ary sequences. The main result shows that the set of all sequences over F_p whose period is a power of p forms a vector space, and that the binomial sequences constitute a basis for it. This supplies a canonical way to express any such periodic sequence as a unique linear combination of the binomial ones.

Core claim

The family of p-ary sequences with period a power of p forms a vector space over F_p, and the family of binomial p-ary sequences is a basis of this space.

What carries the argument

Binomial p-ary sequences, constructed via formation rules that generalize the diagonals of Pascal's triangle modulo p.

If this is right

  • Every sequence with period a power of p admits a unique expression as a linear combination of binomial sequences with coefficients in F_p.
  • The binomial sequences are linearly independent over F_p.
  • The binomial sequences together span the entire vector space of period-p^k sequences.

Where Pith is reading between the lines

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

  • The basis expansion supplies a direct route to computing linear complexity or other invariants by examining the support of the coefficient vector.
  • The same vector-space structure may be used to generate or enumerate all sequences of a given period systematically.

Load-bearing premise

The formation rules for binomial p-ary sequences automatically guarantee linear independence and that they span the full space of period-p^k sequences.

What would settle it

An explicit sequence over F_p with period p^k that cannot be expressed as any linear combination of the binomial sequences, or a nontrivial linear dependence relation satisfied by those sequences.

read the original abstract

The binary binomial sequences correspond to the diagonals of the Pascal's triangle modulo 2. They have interesting properties such as they form a basis of the linear space of all binary sequences with period a power of 2. Other properties of these sequences (period, linear complexity, construction rules or relations among different binomial sequences) have been deeply analysed in detail previously. In this work, we study the binomial $p$-ary sequences for a prime $p$, its intrinsic characteristic and formation rules. We also prove that the family of $p$-ary sequences with period a power of $p$ form a vector space over $\mathbb{F}_p$ and that the family of binomial $p$-ary sequences is a basis of this space.

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 studies binomial p-ary sequences, defined analogously to the binary case as diagonals of Pascal's triangle modulo p. It examines their intrinsic characteristics and formation rules, and proves that the family of all F_p-valued sequences with period a power of p forms a vector space over F_p, with the family of binomial p-ary sequences serving as a basis.

Significance. The generalization of the binary binomial sequence basis property to arbitrary primes is of interest in the theory of periodic sequences over finite fields. The vector-space and basis claims follow from the standard forward-difference argument (linear independence by induction on successive differences), which applies verbatim for any prime p with no additional conditions required. The paper's primary contribution therefore lies in the explicit formation rules and intrinsic characteristics for general p.

minor comments (3)
  1. [Abstract] Abstract: the claim of a proof is stated without any outline of the argument or reference to the forward-difference construction; a one-sentence sketch would improve accessibility.
  2. The definition of the binomial p-ary sequences via formation rules should be stated explicitly (with the precise recurrence or generating function) before the vector-space claim is proved.
  3. Notation for the period-p^m sequences and the indexing of the binomial family should be introduced consistently in the first section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We address the observation regarding the standard nature of the basis argument below.

read point-by-point responses

  1. Referee: The vector-space and basis claims follow from the standard forward-difference argument (linear independence by induction on successive differences), which applies verbatim for any prime p with no additional conditions required. The paper's primary contribution therefore lies in the explicit formation rules and intrinsic characteristics for general p.

    Authors: We agree that the linear independence of the binomial sequences via the forward-difference operator is a standard technique that extends directly to any prime p. The manuscript includes a self-contained presentation of this argument for completeness. The core contribution of the work remains the definition of binomial p-ary sequences, together with their explicit formation rules, intrinsic properties, and relations, which provide a non-trivial generalization of the binary case to arbitrary primes. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central claim is that p-ary sequences of period p^k form a vector space over F_p with the binomial sequences (defined via the natural extension of the binary case, i.e., binom(n,k) mod p) as a basis. This is established by the standard forward-difference argument on the finite rings Z/p^m Z, which shows linear independence and spanning directly from the properties of binomial coefficients and does not reduce to any self-definition, fitted parameter, or self-citation chain. The paper's formation rules coincide with this construction; no load-bearing step collapses to an input by construction. The result is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper adds no free parameters or new entities; it relies on standard finite field arithmetic and linear algebra axioms to establish the basis property.

axioms (1)
  • standard math Vector space axioms hold over finite fields F_p
    The proof that the sequences form a vector space and have a basis relies on these standard properties.

pith-pipeline@v0.9.1-grok · 5650 in / 1167 out tokens · 50178 ms · 2026-06-25T22:17:52.351731+00:00 · methodology

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Reference graph

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