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Second order Recurrences, quadratic number fields and cyclic codes

This paper generalizes the concept of Wall-Sun-Sun primes to primes $p$ where the field $\mathbb{Q}(\sqrt{d})$ is not $p$ -rational, and investigates the weight distributions of cyclic codes over $\mathbb{F}_p$ and $\mathbb{Z}_{p^2}$ associated with these primes, identifying conditions under which they form MDS or NMDS codes.

Original authors: Minjia Shi, Xuan Wang, Bouazzaoui Zakariae, Jon-Lark Kim, Patrick Solé

Published 2026-03-27

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CC BY 4.0

Original authors: Minjia Shi, Xuan Wang, Bouazzaoui Zakariae, Jon-Lark Kim, Patrick Solé

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery involving numbers, patterns, and secret codes. This paper is the case file of a team of mathematicians who are connecting three seemingly unrelated worlds: repeating number patterns, hidden number worlds, and error-correcting codes (the kind that keep your Wi-Fi and space missions working).

Here is the story of their discovery, broken down into simple concepts.

1. The Mystery of the "Double-Period" (The Fibonacci Connection)

Everyone knows the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13... where you add the last two numbers to get the next one.

If you write these numbers down and only look at the "last digit" (mathematicians call this "modulo 10"), the pattern eventually repeats. For example, the sequence of last digits goes 0, 1, 1, 2, 3, 5, 8, 3, 1, 4... and then it starts over. This repeating length is called the period.

The big mystery started with a question: Is there a special prime number (like 2, 3, 5, 7, 11...) where the pattern repeats at the exact same speed whether you look at the last digit (modulo $p$ ) or the last two digits (modulo $p^2$ )?

Analogy: Imagine a clock that ticks every second. Usually, if you look at a clock that ticks every two seconds (a slower version), the pattern changes. But for these special "Wall-Sun-Sun" primes, the fast clock and the slow clock tick in perfect sync.
The Catch: No one has ever found a Wall-Sun-Sun prime for the standard Fibonacci sequence, even though computers have checked up to huge numbers. They are like "unicorns" of the number world.

2. The "Hidden World" of Quadratic Fields

The authors realized that this mystery isn't just about Fibonacci numbers; it's about Quadratic Number Fields.

Analogy: Think of the regular number line as a flat, 2D map. A "Quadratic Number Field" is like a secret, 3D dimension attached to that map. You can only enter this dimension if you have a special key, which is a number like $\sqrt{d}$ (the square root of some number $d$ ).
The Connection: The paper shows that if you find one of those rare "Wall-Sun-Sun" primes, it means the secret 3D dimension (the field $\mathbb{Q}(\sqrt{d})$ ) has a very specific, rare property called being "not p-rational."
The Twist: While we haven't found these primes for the standard Fibonacci sequence ( $d=5$ ), the authors proved that if you change the rules of the sequence slightly (by changing the number $d$ ), you can find infinite examples of these special primes. It's like saying, "We haven't found a unicorn in the forest, but if we look in the desert, there are millions of them!"

3. The Secret Codes (Cyclic Codes)

Now, let's talk about Cyclic Codes. These are the secret languages used to send messages without errors.

Analogy: Imagine you are sending a message to a friend, but you know the line is noisy. You repeat your message in a specific pattern so that if a letter gets garbled, your friend can figure out what it was supposed to be.
The authors took those repeating number patterns (the recurrences) and turned them into these secret codes.
- The "Check Polynomial": This is the rulebook for the code.
- The Result: They found that for these special primes, the codes are incredibly efficient. Some are MDS (Maximum Distance Separable), which is like a code that is as strong as physically possible. Others are NMDS (Near-MDS), which are almost as strong.

4. The "Double-World" Experiment

The most exciting part of the paper is what happens when they compare the codes in two different "worlds":

World A ( $\mathbb{F}_p$ ): A world where numbers wrap around after $p$ (like a clock with $p$ hours).
World B ( $\mathbb{Z}_{p^2}$ ): A slightly more complex world where numbers wrap around after $p^2$ .

Usually, when you move from World A to World B, the behavior of the codes changes completely. It's like taking a song and playing it on a different instrument; it sounds different.

The Discovery:
The authors found that for these special "Wall-Sun-Sun" primes, the weight distribution (a fancy way of counting how many errors the code can catch) in World B is a perfect, predictable "lift" of the pattern in World A.

Analogy: Imagine you have a shadow puppet show (World A). Usually, if you move the light source (World B), the shadow gets distorted. But for these special primes, the shadow in the new world is just a perfect, larger version of the original shadow. The relationship is so tight that if you know the pattern in the small world, you can mathematically predict the pattern in the big world with 100% accuracy.

Why Does This Matter?

For Mathematicians: It connects three huge areas of math (Number Theory, Algebra, and Coding Theory) in a way no one expected. It gives us a new tool to hunt for those elusive "Wall-Sun-Sun" primes.
For Engineers: It helps design better error-correcting codes. If we can build codes that behave predictably even when the math gets complex, we can make our digital communications (5G, satellite TV, deep space probes) more reliable.
The Big Picture: It shows that even in the most abstract corners of mathematics, there are hidden symmetries. Just because a pattern seems random or impossible in one setting doesn't mean it doesn't exist; you just have to look in the right "dimension" (change the $d$ ) to find it.

In a nutshell: The paper is a treasure map. It tells us that if we stop looking for a specific type of treasure in the standard forest, and instead look in the "desert" of different number sequences, we will find an infinite supply of rare primes that make our secret codes work perfectly.

1. Problem Statement

The paper investigates the intersection of three distinct mathematical domains:

Second-order linear recurrences: Specifically, generalizations of the Fibonacci sequence defined by $F_n = -AF_{n-1} - BF_{n-2}$ .
Algebraic Number Theory: Focusing on real quadratic fields $\mathbb{Q}(\sqrt{d})$ and the concept of $p$ -rationality.
Coding Theory: Constructing and analyzing cyclic codes over finite fields $\mathbb{F}_p$ and the ring of integers modulo $p^2$ ( $\mathbb{Z}_{p^2}$ ).

Core Question: The authors explore the condition where a prime $p$ satisfies $k(p) = k(p^2)$ , where $k(m)$ is the period of a specific second-order recurrence modulo $m$ . Primes satisfying this condition for the Fibonacci sequence ( $d=5$ ) are known as Wall-Sun-Sun (WSS) primes. The paper generalizes this to WSS( $d$ ) primes, where the recurrence is tied to the fundamental unit of $\mathbb{Q}(\sqrt{d})$ . The central problem is to characterize these primes, prove the existence of infinitely many such primes for a fixed $p$ (the inverse problem), and determine the weight distributions of the cyclic codes generated by these recurrences.

2. Methodology

The authors employ a multi-faceted approach combining number theory and coding theory:

Theoretical Framework (Number Theory):
- They utilize the definition of $p$ -rational fields: A number field $F$ is $p$ -rational if the Galois group of its maximal pro- $p$ extension (unramified outside $p$ ) has a specific structure related to the $p$ -rank.
- They establish a crucial equivalence (Theorem 1): For a fundamental discriminant $d$ , the field $\mathbb{Q}(\sqrt{d})$ is not $p$ -rational if and only if the period of the associated recurrence satisfies $k(p) = k(p^2)$ . This links the arithmetic property of the field directly to the periodicity of the sequence.
- They use Hensel's Lemma to lift polynomial factorizations from $\mathbb{F}_p[x]$ to $\mathbb{Z}_{p^2}[x]$ , allowing the construction of codes over the ring $\mathbb{Z}_{p^2}$ based on codes over $\mathbb{F}_p$ .
Inverse Problem Construction:
- Instead of searching for $d$ given $p$ (which is computationally hard), the authors prove that for any prime $p > 5$ , there exist infinitely many integers $d$ such that $p$ is a WSS( $d$ ) prime.
- They construct these $d$ by selecting specific polynomials $h(x)$ over $\mathbb{F}_p$ (either reducible or irreducible) with specific orders, lifting them to $\mathbb{Z}_{p^2}$ , and deriving the discriminant $d$ from the coefficients of the lifted polynomial.
Coding Theory Analysis:
- They associate a cyclic code of dimension 2 with the recurrence. The codewords are the period windows of the sequence.
- They analyze the weight distribution (the number of codewords with a specific Hamming weight) for codes over $\mathbb{F}_p$ and $\mathbb{Z}_{p^2}$ .
- They distinguish between MDS (Maximum Distance Separable) codes in the reducible case and NMDS (Near-MDS) codes in the irreducible case.

3. Key Contributions and Results

A. Generalization of WSS Primes

The paper defines WSS( $d$ ) primes as those where the period of the recurrence associated with $\mathbb{Q}(\sqrt{d})$ is identical modulo $p$ and $p^2$ .

Theorem 1: Establishes that $k(p) = k(p^2)$ if and only if $\mathbb{Q}(\sqrt{d})$ is not $p$ -rational.
This provides a new arithmetic characterization of $p$ -rationality via recurrence periods.

B. The Inverse Problem (Existence of $d$ )

The authors prove that for any prime $p > 5$ , there are infinitely many $d$ such that $p$ is a WSS( $d$ ) prime. They categorize the construction into two cases based on the characteristic polynomial:

Reducible Case: The polynomial splits in $\mathbb{F}_p$ . They construct codes with length $n = p-1$ (or divisors thereof).
Irreducible Case: The polynomial is irreducible over $\mathbb{F}_p$ $F_{p}$ . They construct codes with length $n = 2p+2$ $n = 2 p + 2$ (or divisors thereof).
- Proposition 2 & 3: Prove the existence of elements in $\mathbb{F}_{p^2}$ with order $2p+2$ , enabling the construction of irreducible polynomials of this order.

C. Weight Distributions and Code Parameters

The paper provides explicit formulas for the weight distributions of the cyclic codes $C_1$ (over $\mathbb{F}_p$ ) and $C_2$ (over $\mathbb{Z}_{p^2}$ ).

Theorem 2: If $C_1$ $C_{1}$ is a two-weight MDS code with length $n \leq p+1$ $n \leq p + 1$ , the authors derive exact formulas for the number of codewords of weight $n-1$ $n - 1$ and $n$ $n$ for both $C_1$ $C_{1}$ and $C_2$ $C_{2}$ .
- Specifically, $A_{n-1}(C_2) = n(p^2-1)$ and $A_n(C_2) = (p^2-1)(p^2-n+1)$ .
Corollary 1 & 2: They establish divisibility relations between the weight distributions of the field code and the ring code. They also show that under certain conditions (specifically when the code is a repetition of a smaller MDS code), the weight sets of $C_1$ and $C_2$ coincide.

D. Construction of Optimal NMDS Codes

Proposition 5: For primes $p \equiv 3 \pmod 4$ , the authors construct a cyclic code $C_1$ with parameters $[2(p+1), 2, 2p]_p$ .
This code is NMDS (Near-MDS) and has only one non-zero weight.
Significance: The length $2p+2$ is the maximum possible length for a near-MDS code of dimension 2 over $\mathbb{F}_p$ , making these extremal NMDS codes.

E. Computational Verification

The authors used the Magma computer algebra system to generate tables (Tables 1, 2, and 3) listing specific examples of WSS( $d$ ) and WSS( $d$ ) $^*$ primes. These tables provide:

Code lengths ( $n$ ) and discriminants ( $d$ ).
Check polynomials $h(x)$ over $\mathbb{F}_p$ and their Hensel lifts $H(x)$ over $\mathbb{Z}_{p^2}$ .
Weight distributions and classification (MDS, NMDS, 2-MDS, 4-MDS).

4. Significance and Impact

Bridging Number Theory and Coding Theory: The paper creates a robust link between the arithmetic property of $p$ -rationality (a deep concept in Iwasawa theory and class field theory) and the structural properties of cyclic codes. It suggests that coding theoretic properties (like weight distribution) can serve as a proxy for studying difficult number-theoretic conjectures.
Resolution of the Inverse Problem: While the existence of WSS primes for $d=5$ (Fibonacci) remains an open and difficult problem (none are known up to $10^{14}$ ), this paper proves that for any prime $p$ , one can find infinitely many $d$ such that $p$ is a WSS( $d$ ) prime. This shifts the focus from "finding primes" to "constructing fields."
New Families of Optimal Codes: The construction of extremal NMDS codes with length $2p+2$ is a significant contribution to coding theory. These codes are optimal for their length and dimension, and their existence is tied to specific number-theoretic conditions.
Weight Distribution Formulas: The explicit derivation of weight distributions for codes over $\mathbb{Z}_{p^2}$ (a ring, not a field) is technically challenging and provides a template for analyzing codes over other finite commutative rings.

Conclusion

The paper successfully generalizes the concept of Wall-Sun-Sun primes to a broader class of quadratic fields and second-order recurrences. By proving the existence of infinitely many such fields for any given prime and deriving the precise weight distributions of the associated cyclic codes, the authors provide new tools for both algebraic number theorists and coding theorists. The work highlights that the "non-existence" of WSS primes for $d=5$ might be a specific anomaly, whereas the phenomenon is generic for other quadratic fields, offering a fertile ground for future research into the interplay between $p$ -rationality and code optimality.