Good cyclic codes and the uncertainty principle

TL;DR

This paper explores the connection between the uncertainty principle in harmonic analysis and the existence of good cyclic error-correcting codes, suggesting that a weak uncertainty principle could imply their existence.

Contribution

It establishes a novel link between the uncertainty principle and cyclic codes, proposing that a weak form of the principle may guarantee the existence of good cyclic codes.

Findings

A weak uncertainty principle implies the existence of good cyclic codes.

Heuristic arguments support the potential validity of this implication.

The work connects harmonic analysis principles with coding theory challenges.

Abstract

A long standing problem in the area of error correcting codes asks whether there exist good cyclic codes. Most of the known results point in the direction of a negative answer. The uncertainty principle is a classical result of harmonic analysis asserting that given a non-zero function $f$ on some abelian group, either $f$ or its Fourier transform $\hat{f}$ has large support. In this note, we observe a connection between these two subjects. We point out that even a weak version of the uncertainty principle for fields of positive characteristic would imply that good cyclic codes do exist. We also provide some heuristic arguments supporting that this is indeed the case.

Equations

$$(a_0,\dots ,a_{n-1})\in C\iff (a_{n-1},a_0,\dots ,a_{n-2})\in C$$

$$\frac{k}{n}\ge c,\frac{d}{n}\ge c$$

$$|\mathrm{supp}(f)|+|\mathrm{supp}(\hat{f})|\ge p+1,$$

$$(a_0,\dots ,a_{n-1})\mapsto a_0+a_{1}X+\cdots +a_{n-1}{X}^{n-1}$$

$$R={\mathbb{F}}_{\ell }[X]/(X^p-1)\cong {\mathbb{F}}_{\ell }\oplus ({\mathbb{F}}_{{\ell }^r}{)}^s$$

$$R\cong \underset{i=0}{\overset{s}{⨁}}F[X]/(g_i).$$

$$R\cong \underset{i=0}{\overset{p-1}{⨁}}F[X]/(X-{\mu }_i)\cong F^p.$$

$$R\cong {\mathbb{F}}_{\ell }\oplus ({\mathbb{F}}_{{\ell }^r}{)}^s.$$

$$\dim I_f=\dim I_g=p-\mathrm{deg}(g)$$

$$\hat{f}(\chi )=\frac{1}{|A|}\sum _{a\in A}f(a)\overline{\chi (a)}$$

$$(f_1\star f_2)(x)=\frac{1}{|A|}\sum _{a\in A}{f}_1(x-a)f_2(a),$$

$$\widehat{f_1\star f_2}={\hat{f}}_1\cdot {\hat{f}}_2.$$

$$f\mapsto \sum _{a\in A}f(a)a.$$

$$\mathbb{C}[A]⟶{\mathbb{C}}^A$$

$$|\mathrm{supp}(f)|\cdot |\mathrm{supp}(\hat{f})|\ge |A|$$

$$\prod _{\hat{f}(x)\ne 0}\mathbb{C}\subset {\mathbb{C}}^A.$$

$$A=\bigcup _{a\in A}\mathrm{supp}(a\cdot f)\subset \bigcup _{i=1}^r\mathrm{supp}(a_i\cdot f)$$

$$|A|\le \sum _{i=1}^r|\mathrm{supp}(a_i\cdot f)|=r|\mathrm{supp}(f)|=|\mathrm{supp}(\hat{f})|\cdot |\mathrm{supp}(f)|,$$

$$|\mathrm{supp}(f)|+|\mathrm{supp}(\hat{f})|\ge p+1.$$

$$f=a_0+a_{1}X+\cdots +a_{p-1}{X}^{p-1}$$

$$\hat{f}(\xi )=\frac{1}{p}\sum _{i=0}^{p-1}{a}_i{\xi }^{-i}.$$

$$f=\sum _{i=0}^{p-1}{a}_i{X}^i\in \mathbb{C}[X]$$

$$Z(f)\le \mathrm{wt}(f)-1.$$

$$\dim (I_f)=p-Z(f).$$

$$\mathrm{wt}(f)+\dim (I_f)\ge p+1$$

$$f=\prod _{i=1}^{\frac{p-1}{2}}(X-{\xi }^i)\in \mathbb{C}[X].$$

$$\mathrm{wt}(h)\ge p+1-\dim (I_h)\ge p+1-\dim (I_f)=\frac{p+1}{2}$$

$${\mu }_{F,p}(f)=\mathrm{wt}(f)+\dim (I_f).$$

$${\mu }_{F,p}=\min \{{\mu }_{F,p}(f)|0\ne f\in R\}.$$

$$\mathrm{wt}(f)>p-\dim (I_f)=\mathrm{deg}(\gcd (f,X^p-1)).$$

Full text

The uncertainty principle is a classical result of harmonic analysis asserting that given a non-zero function $f$ on some abelian group, either $f$ or its Fourier transform $\hat{f}$ has large support.

In this note, we observe a connection between these two subjects. We point out that even a weak version of the uncertainty principle for fields of positive characteristic would imply that good cyclic codes do exist. We also provide some heuristic arguments supporting that this is indeed the case.

1. Introduction

Let $F$ be a field. Given integers $n$, $k$ and $d$ with $1\leq k\leq n$, an $[n,k,d]_{F}$-code, or code over $F$, is a subspace $C$ of $F^{n}$ of dimension $\dim_{F}(C)=k$, such that for every $0\neq\alpha\in C$, we have $\operatorname{wt}(\alpha)\geq d$, where the weight $\operatorname{wt}(\alpha)$ of a vector $\alpha=(a_{0},\ldots,a_{n-1})\in F^{n}$ is the number of non-zero components $a_{i}$. The integer $d$ is called the distance of the code $C$.

Furthermore, a code $C$ is called cyclic if it is invariant under cyclic permutations of the coordinates, i.e. if

[TABLE]

(see [R, Ch. 8]).

The code $C$, or more properly a family $(C_{n})$ of codes in $F^{n}$ where $n\rightarrow\infty$, possibly along some subsequence of positive integers, is called good if there exists a constant $c>0$ such that

[TABLE]

for all $n$.

We are interested in the case of cyclic codes over a finite field $F$ with $\ell$ elements. The practical interest of such codes goes back at least to Brown and Peterson [BP] (e.g., they can be used to efficiently detect so-called “burst errors”). A long standing open problem in the area of error correcting codes is whether, for a fixed value of $\ell$, there exists an infinite sequence of good cyclic codes.

Most evidence, and maybe the prevailing opinion, goes towards the non-existence of good cyclic codes. Indeed, it was proved by Berman [B] in 1967 that if $\{n\}$ ranges over integers whose prime factors are bounded, and these factors are coprime to the characteristic of the underlying field $\mathbb{F}_{\ell}$, then no sequence of cyclic codes of lengths $\{n\}$, is good. Babai, Shpilka and Stefankovic [BSS] proved that this is also the case if $n$ ranges over integers such that the primes $p$ dividing $n$ all satisfy $p\leq n^{\frac{1}{2}-\epsilon}$ for some fixed constant $\epsilon>0$. Furthermore, they also showed that there are no good cyclic codes that are either locally testable or LDPC (“low density parity check”) codes. We refer to the book [MWS] of MacWilliams and Sloane and to the textbook of Roth [R] for basic terminology and concepts in coding theory.

On the other hand, the uncertainty principle is a classical result of harmonic analysis, which in one form asserts that given a function $f$, either $f$ or its Fourier transform $\hat{f}$ has large support. Many variants exist, and we refer to Folland and Sitaram [FS] for a survey of the continuous setting. We will consider the version of the uncertainly principle where $f:A\rightarrow\mathbb{C}$ is a complex valued function on a finite group $A$, and even more particularly, when $A$ is the cyclic group $\mathbb{Z}/p\mathbb{Z}$ of prime order $p$. In this case, the uncertainty principle states that for $f\not=0$, we have

[TABLE]

where $\operatorname{supp}(g)$ is the support of a function (see Meshulam [M1], Goldstein, Guralnick, Isaacs [GGI], Tao [T] or §3 below).

One can formulate the uncertainty principle for functions from $A=\mathbb{Z}/p\mathbb{Z}$ to any algebraically closed field $F$ (see Section 3). The case of interest to us is when $F$ has positive characteristic $\ell$, in particular when $\ell=2$. The inequality (1.2) does not hold in general in this case (see §4 below), but we will give some heuristic argument suggesting that some weaker version may still hold.

We will then show that even a much weaker version of the inequality (1.2) for $F=\bar{\mathbb{F}}_{2}$ would suffice to imply the existence of good cyclic codes. This should come as quite a surprise, as it goes against the common wisdom in the theory of error correcting codes.

Acknowledgements

The authors are grateful to E. Ben-Sasson, B. Poonen, P. Sarnak and M. Sudan for discussions and suggestions, many of which have been incorporated into the text. We thanl F. Voloch for pointing out his note [V]. We acknowledge support by the ERC, NSF, ISF, Dr. Max Rössler, the Walter Haefner Foundation and the ETH Foundation, and the ETH Institute for Theoretical Studies. EK’s work is partially supported by an DFG-SNF lead agency program grant (grant 200021L_153647).

1.1. Organization of the paper

This note is arranged as follows:

In $\S$ 2, we describe cyclic codes of length $n$ over the prime field $\mathbb{F}_{\ell}$ of order $\ell$, as ideals in the group algebra $\mathbb{F}_{\ell}[\mathbb{Z}/n\mathbb{Z}]\cong\mathbb{F}_{\ell}[x]/(x^{n}-1)$. We then describe the structure and the ideals of $\mathbb{F}_{\ell}[\mathbb{Z}/p\mathbb{Z}]$ when $n=p$ is a prime, and express the dimension and the distance of such an ideal in terms of this data (using in particular the multiplicative order of $\ell$ modulo $p$).

In $\S$ 3, we formulate the uncertainty principle for functions $f:\mathbb{Z}/p\mathbb{Z}\rightarrow\mathbb{C}$. To illustrate the connection with cyclic codes, we show how this uncertainty principle implies the existence of good cyclic codes over $\mathbb{C}$ – the examples we recover are the well-known Reed-Solomon codes over $\mathbb{C}$. This is of course not the end of the story, as one wants such codes over finite fields.

In $\S$ 4, we formulate a few variants of the uncertainty principle over various fields. We present a proof of the uncertainty principle for any field of characteristic zero, following [GGI]. Afterwards, we present some counter-examples to a naive generalization of the uncertainty principle to finite fields.

In $\S$ 5, we propose a weaker version of uncertainty principle, and show how this weaker version implies the existence of good cyclic codes. In $\S$ 6, we present some heuristics, both for this weak uncertainty principle and for the existence of good cyclic codes.

We conclude with an Appendix that explains that the uncertainty principle for $\mathbb{Z}/p\mathbb{Z}$ is equivalent to an old result of Chebotarev.

2. Cyclic codes

2.1. Introduction

The following is a long standing open problem.

Problem 2.1**.**

Are there good cyclic codes over a fixed finite field $F$?

This was asked by MacWilliams and Sloane [MWS, Problem 9.2, p. 270]. See also [MPW] who attribute the problem to [AMS]. It seems that the common belief is that there are no such codes and there are a number of results in support of such a conjecture.

For instance, the most commonly used cyclic codes are the long BCH codes (see [R, §5.6] for definition and background of BCH codes), and Lin and Weldon [LW] proved that these codes are not good.

Partial results toward the conjecture were obtained by Berman [B] in 1967 and by Babai, Shpilka and Stefankovic [BSS] in 2005. We state their results formally:

Theorem 2.2** (Berman).**

Let $F$ be a finite field of order $\ell$, and $(C_{t})_{t}$ a family of $[n_{t},k_{t},d_{t}]_{F}$-cyclic codes such that there exists some real number $c>0$ with $\frac{k_{t}}{n_{t}}\geq c$ for all $t$. Assume furthermore that there exists $\beta\geq 1$ such that all primes dividing $n_{t}$ are coprime to $\ell$ and at most $\beta$. Then there exists an integer $m$, depending on $\ell$ and $\beta$, such that $d_{t}\leq m$. In particular, this family is not a good family of codes.

Theorem 2.3** (Babai-Shpilka-Stefankovic).**

Let $F$ be a finite field, and let $(C_{t})_{t}$ be a family of $[n_{t},k_{t},d_{t}]_{F}$-cyclic codes over $F$. Assume that there exists $\delta>0$, independent of $t$, such that for every $t$ and for every prime $p$ dividing $n_{t}$, we have $p<n_{t}^{1/2-\delta}$. Then the family $(C_{t})_{t}$ is not a good family of codes over $F$.

There are other results which give some support to a negative answer to Problem 2.1, for example:

Theorem 2.4** (Babai-Shpilka-Stefankovic).**

Let $F$ be a finite field. Then:

• •

There are no good cyclic LDPC (low density parity check) codes over $F$;

• •

There are no good cyclic locally testable codes over $F$.

We refer to [McK, Ch. 47] for the definition of LDPC codes, and to [GS] for locally testable codes; these are important concepts in coding theory in recent years.

Let $F$ be any field. The key to the investigation of cyclic codes over $F$ is their description in algebraic terms using the polynomial ring $F[X]$.

Proposition 2.5**.**

Let $n\geq 1$ be an integer. Under the isomorphism

[TABLE]

between $F^{n}$ and the ring $R=F[X]/(X^{n}-1)$, a subspace $C\subset R$ is a cyclic code over $F$ if and only if $C$ is an ideal of $R$.

Proof.

Indeed, an $F$-vector subspace of $R$ is a cyclic code if and only if $XP\in C$ for any $P\in C$, which is equivalent to asking that $C$ be an ideal of $R$. ∎

It will also often be convenient to identify the ring $R$ with the subspace of polynomials $P\in F[X]$ of degree less than $n$.

2.2. Describing the ideals of $R=F[X]/(X^{n}-1)$

If we specialize to the case where $n=p$ is a prime number, we can describe $R$ and its ideals in quite concrete and well-known terms:

Proposition 2.6**.**

Let $p$ be a prime number different from the characteristic $\operatorname{char}(F)$ of $F$. Then:

1. (1)

The ring $R=F[X]/(X^{p}-1)$ is a direct sum of finite extensions of $F$; these finite extensions are in one to one correspondence with the irreducible factors of the polynomial $X^{p}-1\in F[X]$. 2. (2)

If $x^{p}-1$ splits in linear factors in $F[x]$ (e.g. if $F$ is algebraically closed), then $R$ is isomorphic to $F^{p}$ as a ring; 3. (3)

Assume that $F=\mathbb{F}_{\ell}$ is a finite field of order $\ell$. Let $r=\operatorname{ord}_{p}(\ell)$, i.e. the order of $\ell$ as an element of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{*}=\mathbb{F}_{p}^{*}$. Denote $s=(p-1)/r$. Then

[TABLE]

i.e., it is isomorphic as a ring to a direct sum of $\mathbb{F}_{\ell}$ and $s$ copies of the extension $\mathbb{F}_{\ell^{r}}$ of $\mathbb{F}_{\ell}$.

Proof.

1. (1)

As $p\neq\operatorname{char}(F)$, the polynomial $X^{p}-1$ is separable in $F[X]$ and hence factors as a product of distinct irreducible polynomials $\prod_{i=0}^{s}g_{i}$, where we put $g_{0}=X-1$. It then follows from the Chinese Remainder Theorem that

[TABLE]

Since $g_{i}$ is irreducible, each quotient ring $F[X]/(g_{i})$ is a field extension of $F$ of degree $\deg(g_{i})$. 2. (2)

By assumption, $X^{p}-1=\prod_{i=0}^{p-1}(X-\mu_{i})$, where $\mu_{i}$ runs over the $p$-th roots of unity in $F$. Since $F[X]/(X-\alpha)\cong F$, we get an isomorphism

[TABLE] 3. (3)

Since $\mathbb{F}_{p}^{*}$ is a cyclic group of order $p-1$, the order $r$ of $\ell$ modulo $p$ divides $p-1$, and hence $s=(p-1)/r$ is an integer.

We have $\ell^{r}\equiv 1(\mbox{mod }p)$ and $\mathbb{F}_{\ell^{r}}^{*}$ is a cyclic group of order $\ell^{r}-1$, hence the field extension $\mathbb{F}_{\ell^{r}}$ of $\mathbb{F}_{\ell}$ contains an element of order $p$, and is the smallest extension with this property. In fact, the field $\mathbb{F}_{\ell^{r}}$ contains all the $p$-th roots of unity, i.e. $\mathbb{F}_{\ell^{r}}$ is the splitting field of the polynomial $X^{p}-1$. For every $p$-th root of unity $\mu$, the extension $\mathbb{F}_{\ell}[\mu]$ is equal to $\mathbb{F}_{\ell^{r}}$ (in a fixed algebraic closure of $\mathbb{F}_{\ell}$). This shows that all the irreducible factors $g_{i}$ of $X^{p}-1$, with the exception of $X-1$, are of degree $r$. Hence

[TABLE]

∎

We can now describe the ideals of $R$. Since $R$ is a direct sum of fields, every ideal in $R$ is the direct sum of a certain subset of these fields. If $F$ is algebraically closed, for instance, we see that $R$ has $\binom{p}{i}$ distinct ideals of dimension $i$, for every $0\leq i\leq p$, and a total of $2^{p}$ ideals.

If $F=\mathbb{F}_{\ell}$ where $\ell$ is the power of a prime number, let $r$ be the order of $\ell$ modulo $p$ and $s=\frac{p-1}{r}$ as in the proposition. In the special case $r=1$, namely when $p\mid\ell-1$, the polynomial $X^{p}-1$ splits completely in $\mathbb{F}_{\ell}[X]$ and the ideals are exactly the same as those in the algebraically closed case.

Now assume that $r>1$, which is the case we are most interested in since we will consider a fixed value of $\ell$ as $p$ tends to $\infty$. Then $R$ has $\binom{s}{i}$ ideals of dimension $ir$ and $\binom{s}{i}$ ideals of dimension $ir+1$ for all integers $i$ with $0\leq i\leq s$. Hence the total number of ideals in $R$ is $2^{s+1}$.

We note that $r\geq\log_{\ell}(p+1)$, and hence $s\leq\frac{p-1}{\log_{\ell}(p+1)}$.

There are two extreme cases which are worth singling out, although whether they actually occur is somewhat conjectural:

**(a): **

Assume that $\ell$ is a primitive root mod $p$, i.e. $\ell$ generates the cyclic group $(\mathbb{Z}/p\mathbb{Z})^{*}$. Then $r=p-1$ and so $s=1$, i.e. $R\cong\mathbb{F}_{l}\oplus\mathbb{F}_{l^{p-1}}$ and $R$ has only two non-trivial ideals.

**(b): **

Assume that $\ell=2$ and that $p$ is a Mersenne prime, namely $p=2^{m}-1$ for some $m\geq 2$. Then we have $r=m=\log_{2}(p+1)$ and $s=\frac{p-1}{\log_{2}(p+1)}$; in this case, $R$ has the “maximal” possible number of ideals $2^{\frac{p-1}{\log_{2}(p+1)}+1}$.

We stated that it is not known if these cases occur infinitely often. Indeed, it is a very famous conjecture of Artin (see Moree’s survey [Mo]) that, for a given prime number $\ell$, there exist infinitely many primes $p$ such that $\ell$ is a primitive root modulo $p$. The validity of this conjecture is extremely likely, since it was shown by Hooley [H] to follow from a suitable form of the Generalized Riemann Hypothesis. Moreover, although it not known to hold for any concrete single prime $\ell$, Heath-Brown [HB] has shown that it holds for all but at most two (unspecified) prime numbers.

On the other hand, although it is expected that there are infinitely many Mersenne primes, very little is known about this question, or about small values of $\operatorname{ord}_{p}(2)$ in general, even assuming such conjectures as the Generalized Riemann Hypothesis (see however Lemma 6.2).

The most convenient analytic criterion to find primes with $\operatorname{ord}_{p}(\ell)$ under control is the following elementary fact:

Lemma 2.7**.**

Let $\ell$, $q$ and $p$ be different primes. If $p$ is totally split in the extension $K_{q,\ell}=\mathbb{Q}(e^{2i\pi/q},\sqrt[q]{\ell})$, then $p$ is congruent to $1$ modulo $q$ and the order of $\ell$ modulo $p$ divides $(p-1)/q$, in particular $\operatorname{ord}_{p}(\ell)<p/q$.

Proof.

Let $\mathcal{O}$ be the ring of integers of $K_{q,\ell}$. If $p$ is totally split in $K_{q,\ell}$, then the quotient ring $\mathcal{O}/p\mathcal{O}$ is a product of copies of the field $\mathbb{F}_{p}$. So $\mathbb{F}_{p}$ contains the $q$-th roots of unity (in particular, $q\mid p-1$) and the $q$-th roots of $\ell$. So $\ell$ is an $q$-th power in $\mathbb{F}_{p}$, which means that $\operatorname{ord}_{p}(\ell)$ divides $(p-1)/q$. ∎

Note that as an application of Chebotarev’s density Theorem [N, Th. 13.4], for any primes $q,\ell$, there exists infinitely many primes which totally splits in $K_{q,\ell}$.

To summarize the discussion: the ideals of $R$ and their dimensions can be easily described, although the existence of certain configurations might be subject to the truth of certain arithmetic conjectures.

It is more complicated to evaluate the distance of ideals of $R$ when interpreted as cyclic codes. For this we will use the Fourier transform and the uncertainty principle in the next section. We begin first with a general lemma.

Lemma 2.8**.**

Let $p$ be a prime. For any polynomial $f\in F[X]$, let $I_{f}$ be the ideal generated by the image of $f$ in $R=F[X]/(X^{p}-1)$ and let $g=\gcd(f,X^{p}-1)$.

1. (1)

We have $I_{f}=I_{g}$, i.e. the ideal generated by $f$ is the same as the ideal generated by the greatest common divisor of $f$ and $X^{p}-1$. 2. (2)

We have

[TABLE]

Proof.

(a) We obviously have $\gcd(f,X^{p}-1)\mid f$ in $F[X]$, and since $F[X]$ is a principal ideal domain, there exist polynomials $h_{1}$ and $h_{2}$ in $F[X]$ such that $\gcd(f,X^{p}-1)=h_{1}f+h_{2}(X^{p}-1)$. Hence we get $f\mid\gcd(f,X^{p}-1)$ in $R$, which proves claim (a).

(b) The first equality follows from (a). For the second equality, it suffices to note that, by euclidean division by the polynomial $(X^{p}-1)/g$ of degree $d=p-\deg(g)$, the elements $\{X^{i}\cdot f\,|\,i=0,1,\ldots,d-1\}$ form a basis of $I_{f}$. ∎

For later reference, we will denote $\operatorname{Z}(f)=\deg(\gcd(f,X^{p}-1))$ for any polynomial $f\in F[X]$ and any prime $p$. If $F$ has characteristic different from $p$, then $X^{p}-1$ is a separable polynomial, and in that case, the integer $\operatorname{Z}(f)$ is therefore the number of $p$-th roots of unity $\xi$, in an algebraic closure of $F$, such that $f(\xi)=0$. This interpretation will be very useful as we now turn to the uncertainty principle…

3. The uncertainty principle over $\mathbb{C}$

3.1. The Fourier transform on finite abelian groups

Let $A$ be a finite abelian group. The dual group $\widehat{A}$ of $A$ is the group of all homomorphisms $A\to\mathbb{S}^{1}$, where $\mathbb{S}^{1}$ is the group of complex numbers of modulus $1$. The product on $\widehat{A}$ is the pointwise multiplication of functions. The dual group is also a finite abelian group, in fact it is isomorphic to $A$ (non-canonically).

The Fourier transform on $A$ is a linear map from the space $L^{2}(A)=\mathbb{C}^{A}$ of complex-valued functions on $A$ to the analogue space $L^{2}(\widehat{A})$ of complex-valued functions on the dual group. For a function $f\colon A\to\mathbb{C}$, its Fourier transform $\widehat{f}\colon\widehat{A}\to\mathbb{C}$ is defined by

[TABLE]

for any $\chi\in\widehat{A}$.

The Fourier transform is also an algebra isomorphism, where $L^{2}(A)$ is viewed as an algebra with the convolution product

[TABLE]

and $L^{2}(\widehat{A})$ has the pointwise product of functions. In other words, we have

[TABLE]

The connection that we will make with cyclic codes emphasizes the group algebra of a cyclic group. It is therefore convenient to interpret the Fourier transform in terms of the group algebra $\mathbb{C}[A]$ of the group $A$ instead of $L^{2}(A)$.

We identify $L^{2}(A)$ and $\mathbb{C}[A]$ by the map

[TABLE]

Then the Fourier transform gives an isomorphism

[TABLE]

of algebras over $\mathbb{C}$, where the image of the standard basis $\{a\in A\}$ is the basis of characters of the algebra of functions $\mathbb{C}^{|A|}$.

3.2. The general uncertainty principle for finite abelian

groups

For $f\in L^{2}(A)$, or equivalently $f\in\mathbb{C}[A]$, we denote by $\operatorname{supp}(f)$ the support of $f$, namely the set of $a\in A$ such that $f(a)\neq 0$.

Intuitively, by“uncertainty principle”, we mean a statement that asserts that there are no non-zero functions $f$ such that both $f$ and $\widehat{f}$ have “small” support (for instance, in the continuous case, there is no non-zero smooth function with compact support whose Fourier transform is also compactly supported). There are many variants of this principle. One well-known elementary “uncertainty principle” version, valid for all finite abelian groups, is the following result of Donoho and Stark [DS, §2]:

Proposition 3.1** (Uncertainty principle).**

Let $A$ be a finite abelian group and let $f\not=0$ be a function from $A$ to $\mathbb{C}$. Then we have

[TABLE]

We present the proof of this fact from [GGI], which fits well with our point of view of working with group algebras. For other proofs and generalizations, we refer to the papers [M2], [M3] and [T], as well as to the references contained in those articles.

Proof.

We view $f$ as an element of the group algebra $\mathbb{C}[A]$, which is commutative. Let $I=(f)$ be the principal ideal generated by $f$. Using the isomorphism $\mathbb{C}[A]\simeq\mathbb{C}^{A}$ given by the Fourier transform, as we recalled above, the ideal $I$ corresponds to the principal ideal in $\mathbb{C}^{A}$ generated by the Fourier transform of $f$. This ideal is simply

[TABLE]

In particular, the dimension $r$ of $I$, as a $\mathbb{C}$-vector space, is the cardinality of the support of $\widehat{f}$. Since the elements $a\cdot f$ for $a\in A$ span $I$ as $\mathbb{C}$-vector space, there exist $r$ elements $a_{1}$, …, $a_{r}$ such that $I$ is the span of $a_{1}\cdot f$, …, $a_{r}\cdot f$.

For any $a\in A\subset\mathbb{C}[A]$, the support of $a\cdot f$ is $a\cdot\operatorname{supp}(f)$. Since $f\neq 0$, its support is not empty, hence for any $x\in A$, we can find some element $a\in A\subset\mathbb{C}[A]$ such that $x\in\operatorname{supp}(a\cdot f)$.

We then have

[TABLE]

which implies that

[TABLE]

as claimed. ∎

3.3. The uncertainty principle for simple cyclic groups

In the late 1980’s, R. Meshulam observed that an old result of Chebotarev implies a version of the uncertainty principle for cyclic groups of prime order $p$ that is much stronger than Proposition 3.1. This strong version has been rediscovered several times since then, and admits a number of proofs and generalizations (see for instance, Chebotarev [C], Meshulam [M1, M2, M3], Goldstein, Guralnick and Isaacs [GGI], Tao [T], Stevenhagen and Lenstra [SL], and the references therein).

Theorem 3.2** **(Uncertainty principle for cyclic groups

of prime order).

Let $A$ be a cyclic group of prime order $p$, and $f\not=0$ an element of $\mathbb{C}[A]$. Then

[TABLE]

We will postpone the proof to Section 3.2, and in the appendix, we will also explain Meshulam’s original observation that this statement is equivalent to a classical result of Chebotarev about Vandermonde matrices.

To bring the connection with codes, we will now reformulate this statement. The group algebra $\mathbb{C}[\mathbb{Z}/p\mathbb{Z}]$ of the cyclic group of order $p$ is isomorphic to the quotient algebra $R=\mathbb{C}[X]/(X^{p}-1)$ by mapping the generator $1$ of $\mathbb{Z}/p\mathbb{Z}$ to the image of $X$. The dual group $\widehat{\mathbb{Z}/p\mathbb{Z}}$ is isomorphic to the group $\mu_{p}(\mathbb{C})$ of $p$-th roots of unity in $\mathbb{C}$, by mapping a character $\chi$ to the $p$-th root of unity $\chi(1)$. The Fourier transform of an element $f\in R$, represented as the image of a polynomial

[TABLE]

is then identified with the function defined on $p$-th roots of unity by

[TABLE]

In other words, $\widehat{f}$ is the evaluation of the representing polynomial (3.3) at roots of unity.

With this notation, recalling the definition $\operatorname{Z}(f)=\deg(\gcd(f,X^{p}-1))$ and the fact that this is number of zeros of $f$ among $p$-th roots of unity, the uncertainty principle of Theorem 3.2 gets the following form:

Theorem 3.3**.**

Let $p$ be a prime. For any polynomial

[TABLE]

of degree $<p$, let $\operatorname{wt}(f)=|\{i|a_{i}\neq 0\}|$ and let $\operatorname{Z}(f)=|\{\mu\in\mu_{p}(\mathbb{C})|f(\mu)=0\}|$, i.e. the number of $p$-th roots of unity of $f$ which are also roots of $f$. Then we have

[TABLE]

Indeed, by definition, if we view $f$ as an element of $R=\mathbb{C}[\mathbb{Z}/p\mathbb{Z}]$, then we have $|\operatorname{supp}(f)|=\operatorname{wt}(f)$ and $|\operatorname{supp}(\widehat{f})|=p-\operatorname{Z}(f)$, and therefore (3.2) and (3.4) are equivalent.

Remark 3.4**.**

(1) The restriction $\deg(f)<p$ is necessary: the polynomial $f=X^{p}-1$ has $\operatorname{wt}(f)=2$ and $\operatorname{Z}(f)=p$.

(2) The inequality (3.4) is best possible. For instance, the cyclotomic polynomial $f=\frac{X^{p}-1}{X-1}=1+X+\ldots+X^{p-1}$ vanishes on all the non-trivial $p$-roots of unity, so $\operatorname{Z}(f)=p-1=\operatorname{wt}(f)-1$. Another example is $f=X-1$, in which case we also obtain $\operatorname{Z}(f)=1=\operatorname{wt}(f)-1$.

We can now use Lemma 2.8 to obtain another reformulation of Theorems 3.2 and 3.3. The point is that if $f$ is a polynomial in $\mathbb{C}[X]$ of degree $<p$, viewed also as an element of $R$, then by Lemma 2.8 (2), the dimension of the ideal $I_{f}$ generated by the image of $f$ in $R$ satisfies

[TABLE]

From Theorem 3.3, we get therefore:

Theorem 3.5** (Uncertainty principle reformulated).**

For every non-zero polynomial $f\in\mathbb{C}[X]$ of degree $<p$, considered as an element of $R=\mathbb{C}[X]/(X^{p}-1)$, we have:

[TABLE]

when $I_{f}=(f)$ is the ideal of $R$ generated by the image of $f$.

We conclude this section by showing how this interpretation of the uncertainty principle gives a good family of cyclic codes over $\mathbb{C}$:

Corollary 3.6**.**

There exists a family of good cyclic codes over $\mathbb{C}$.

Proof.

Let $\xi=e^{\frac{2\pi i}{p}}\in\mathbb{C}$, and define

[TABLE]

Since $f|(X^{p}-1)$, we have $\dim(I_{f})=p-\deg(f)=\frac{p+1}{2}$ by Lemma 2.8 (2).

Let then $h\not=0$ be an element of $I_{f}$. We then have $\dim(I_{h})\leq\dim(I_{f})$, so that

[TABLE]

by Theorem 3.5. The ideal $C_{p}=I_{f}$ is therefore a $[p,\frac{p+1}{2},\frac{p+1}{2}]_{\mathbb{C}}$-cyclic code, and the family $\{C_{p}\}_{p\text{ prime}}$ is a good family of cyclic codes. ∎

The codes we have “found” in this proof are special cases of the famous Reed-Solomon codes (see, e.g., [R, §5.2]).

4. Uncertainty principle for general fields

4.1. General statements

The formulation of the uncertainty principle in Theorems 3.3, in the form of the inequality (3.4) and in Theorem 3.5, through (3.5), make sense for all fields. As we will see later, these statements are not true in such generality, but they might be true, and useful, in some weaker form. For this reason, we make the following definition.

Definition 4.1**.**

Let $F$ be a field, $p$ a prime number and $R=F[X]/(X^{p}-1)$. For $f\in R$, represented by a polynomial of degree $<p$, we denote by $I_{f}$ the ideal generated by $f$ in $R$, and we denote

[TABLE]

We then define the invariant

[TABLE]

We will sometimes write $\mu(f)$ instead of $\mu_{F,p}(f)$, when the field and prime involved are clear in context.

Here are some simple observations:

• •

If $E/F$ is a field extension and $f\in F[X]/(X^{p}-1)$, then $\mu_{F,p}(f)=\mu_{E,p}(f)$ for any prime number $p$. In particular, it follows that $\mu_{E,p}\leq\mu_{F,p}$ for each $p$.

• •

For $f=1+X+\ldots+X^{p-1}$, we have $\operatorname{wt}(f)=p$ and $\dim(I_{f})=1$. It follows that $\mu_{F,p}\leq p+1$ for any field $F$ and any prime $p$.

• •

According to the uncertainty principle for $F=\mathbb{C}$ (Theorems 3.2, 3.3 and 3.5), we have $\mu_{\mathbb{C},p}=p+1$ for every prime $p$.

So for any field we can state the uncertainty principle as follows:

Definition 4.2** (Uncertainty principle).**

A field $F$ is said to satisfy the uncertainty principle if, for any prime number $p$, we have $\mu_{F,p}>p$, or equivalently if $\mu_{F,p}=p+1$, for all $p$.

As we shall see in §4.2, the uncertainty principle does not hold in general, but let us start with some positive results:

Proposition 4.3**.**

Let $F=\mathbb{F}_{\ell}$ be the finite field of prime order $\ell$ and assume that $\ell$ is a primitive root modulo $p$, i.e., that $\operatorname{ord}_{p}(\ell)=p-1$. Then $\mu_{F,p}=p+1$.

Proof.

Let $\xi\neq 1$ be a primitive $p$-th root of unity in $\bar{\mathbb{F}}_{\ell}$. As recalled in Section 2.2, the extension $\mathbb{F}_{\ell}(\xi)/\mathbb{F}_{\ell}$ is then of degree $\operatorname{ord}_{p}(\ell)=p-1$. This implies that the polynomial $\frac{X^{p}-1}{X-1}=1+X+\ldots+X^{p-1}$ is irreducible over $\mathbb{F}_{\ell}$. In particular, for every polynomial $f\in\mathbb{F}_{\ell}[X]$ of degree less then $p$, the gcd of $f$ and $X^{p}-1$ can only be one of $1$, $X-1$ or $(X^{p}-1)/(X-1)$. Then the dimension $\dim(I_{f})=p-\deg(\gcd(f,X^{p}-1))$ is equal to $p$, $p-1$ or $1$, respectively (Lemma 2.8 (2)).

We consider each case in turn and show that $\mu(f)\geq p+1$ in any case. If $\dim(I_{f})=p$, then since $\operatorname{wt}(f)\geq 1$ (because $f\neq 0$), we get $\mu(f)\geq p+1$. If $\dim(I_{f})=p-1$, then we have $\gcd(f,X^{p}-1)=X-1$, so $X-1\mid f$. Since the only non-zero polynomials of weight $1$ are monomials $cX^{i}$ with $c\not=0$, and $X-1\nmid cX^{i}$ for $0\leq i<p$, we must have $\operatorname{wt}(f)\geq 2$, and therefore $\mu(f)\geq p-1+2=p+1$. Finally, if $\dim(I_{f})=1$, then we have $f=c\sum_{i=0}^{p-1}X^{i}$ for some $c\not=0$, and then $\operatorname{wt}(f)=p$ and $\mu(f)=p+1$. ∎

Another case is the following claim (which appears also in [F, Lemma 2] and [GGI, Lemma 6.5]), that we will use later:

Proposition 4.4**.**

Let $p$ be a prime and let $F$ be a field of characteristic $p$. Then we have $\mu_{F,p}=p+1$.

Proof.

By Lemma 2.8 (2), we need to show that for any $0\neq f\in F[X]/(X^{p}-1)$, we have

[TABLE]

Since $F$ has characteristic $p$, we have $X^{p}-1=(X-1)^{p}$, which means that there exists some integer $m$ with $0\leq m<p$ such that $\gcd(f,X^{p}-1)=(X-1)^{m}$. So we need to prove that for a polynomial $f$ with $(X-1)^{m}|f$, we have $\operatorname{wt}(f)>m$.

We proceed by induction on $\deg(f)<p$. In the base case $\deg(f)=0$, we have $f=c\neq 0$. Then $X-1\nmid f$, so that $m=0$ and $\operatorname{wt}(f)=1>m$, as claimed.

Now assume that the property is valid for all polynomials of degree $<\deg(f)$ and that $(X-1)^{m}|f$. If $f(0)=0$, we deduce that $(X-1)^{m}|f(X)/X$, hence by induction we obtain $m<\operatorname{wt}(f/X)=\operatorname{wt}(f)$. If $f(0)\neq 0$, on the other hand, then we consider the derivative $f^{\prime}$ of $f$. From $(X-1)^{m}\mid f$, it follows that $(X-1)^{m-1}\mid f^{\prime}$: indeed, writing $f=f_{1}(X-1)^{m}$ and differentiating, we get $f^{\prime}=f^{\prime}_{1}(X-1)^{m}+mf_{1}(X-1)^{m-1}$, which is divisible by $(X-1)^{m-1}$. By induction, we therefore get $\operatorname{wt}(f^{\prime})>m-1$. But then, since $f(0)\neq 0$ and $m<p$, we have $\operatorname{wt}(f)=\operatorname{wt}(f^{\prime})+1>m$, as needed. ∎

4.2. Fields of characteristic zero

We will now present a proof (following [GGI]) of the uncertainty principle for any field $F$ of characteristic zero. Note that Theorems 3.2, 3.3 and 3.5 are special cases of this result, where the field is $\mathbb{C}$. Since it is elementary that we need only prove the uncertainty principle for finitely generated fields $F$, and since such a field $F$ of characteristic [math] can be embedded into $\mathbb{C}$, we could simply deduce the result from the case of $\mathbb{C}$. We give a complete proof anyway.

The next lemma is the key step in the proof.

Lemma 4.5** (Specialization).**

Let $p$ be a prime, $F$ a field of characteristic [math], and

[TABLE]

a non-zero element of $R=F[X]/(X^{p}-1)$. Then for every prime number $q$, there exists a field $E$ of characteristic $q$ and a polynomial $\tilde{f}\in E[X]/(X^{p}-1)$ such that $\operatorname{wt}(\tilde{f})\leq\operatorname{wt}(f)$ and $\dim_{E}(I_{\tilde{f}})\leq\dim_{F}(I_{f})$.

Sketch of the proof:.

c

1. (1)

Since $\operatorname{char}(F)=0$, the field $\mathbb{Q}$ is a subfield of $F$. Let $A=\mathbb{Q}[a_{0},\ldots,a_{p-1}]$, which is a $\mathbb{Q}$-subalgebra of $F$. By Hilbert’s Nullstellensatz, the homomorphisms $\phi\colon A\to\bar{\mathbb{Q}}$, where $\bar{\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$, separate the points of $A$, and therefore there exists a morphism $\phi:A\rightarrow\bar{\mathbb{Q}}$, such that $\phi(a_{i})\neq 0$ for every $i$, with $0\leq i\leq p-1$, such that $a_{i}\not=0$. Let $K_{1}$ be the number field (a finite extension of $\mathbb{Q}$) generated by the image of $\phi$ and $f_{1}$ the polynomial

[TABLE]

Then by the definition of $K_{1}$, we have $\operatorname{wt}(f_{1})=\operatorname{wt}(f)$. Moreover, $\phi$ induces an isomorphism between the $p$-th roots of unity in $\bar{K}$ and those in $\bar{\mathbb{Q}}$, so that $\operatorname{Z}(f)=\operatorname{Z}(f_{1})$ also. This means that we may replace $K$ and $f$ by $K_{1}$ and $f_{1}$, and reduce to the case where $K$ is a number field. 2. (2)

Let $\mathcal{O}_{K}$ be the ring of integers of $K$, and $\mathfrak{m}$ a maximal ideal in $\mathcal{O}_{K}$ that contains $q\in\mathbb{Z}\subset\mathcal{O}_{K}$. Then $E=\mathcal{O}_{K}/\mathfrak{m}$ is a finite field of characteristic $q$. 3. (3)

Let $t\in\mathcal{O}_{K}$ be a non-zero integer such that $ta_{i}\in\mathcal{O}_{K}$ for all $i$, and such that there exists some $i$ such that $ta_{i}\notin\mathfrak{m}$ (this exists because not all $a_{i}$ are zero). Then, if $\tilde{f}$ is the image of $tf$ under the reduction map from $\mathcal{O}_{K}$ to $E$, we have $\tilde{f}\neq 0$ in $E[X]$, and $\tilde{f}$ is a polynomial of degree $<p$. 4. (4)

By construction, we have $\operatorname{wt}(\tilde{f})\leq\operatorname{wt}(f)$. On the other hand, we get

[TABLE]

∎

Theorem 4.6**.**

For every field $F$ of characteristic [math] and every prime $p$, we have $\mu_{F,p}=p+1$, i.e., the uncertainty principle is true over any field of characteristic [math].

Proof.

Let $F$ be a field of characteristic zero, and let $p$ be a prime. Let $f\in F[X]/(X^{p}-1)$ be non-zero. By the Specialization Lemma 4.5 with $q=p$, there exists a field $E$ of characteristic $p$ and a non-zero element $\tilde{f}\in E[X]/(X^{p}-1)$ such that $\mu_{E,p}(\tilde{f})\leq\mu_{F,p}(f)$. Because $E$ has characteristic $p$, Proposition 4.4 implies that $\mu_{F,p}(f)\geq\mu_{E,p}(\tilde{f})>p$. Since this holds for all $f$, the result follows. ∎

4.3. Counter examples to the uncertainty principle over finite

fields

Specific examples of finite fields $F$ for which the uncertainty principle of Definition 4.2 does not hold over a finite field $F$ are given in [GGI]. One such example is $F=\mathbb{F}_{2}$. If we take $p=7$ and $f=X^{3}+X+1\in\mathbb{F}_{2}[X]/(X^{7}-1)$, then we have

[TABLE]

hence $\dim(I_{f})=4$ while $\operatorname{wt}(f)=3$, so that $\mu_{\mathbb{F}_{2},7}\leq 7$.

The next counter-examples to the naive uncertainty principal for finite fields were suggested to us by Madhu Sudan.

Let $q<p$ be two different primes, and $r=\operatorname{ord}_{p}(q)$. Let $F=\mathbb{F}_{q}$ and $E=\mathbb{F}_{q^{r}}$, so that $E$ contains all the $p$-th roots of unity. Moreover, $E$ is generated as an $F$-vector space by the $p$-th roots of unity. We consider the trace polynomial

[TABLE]

A basic but crucial observation is that the function from $E$ to $E$ defined by the trace polynomial $T$ is a surjective $F$-linear map from $E$ to the subfield $F$, which we denote $\mathrm{tr}$. In particular, $\mathrm{tr}$ is not identically zero on $E$, and since the $p$-th roots of unity generate $E$ as $F$-vector space, this means that $T$ is not identically zero on the $p$-th roots of unity.

By the pigeon-hole principle, there exists some $\alpha\in F$ such that at least $\frac{p}{q}$ of the $p$-th roots of unity in $E$ are roots of $T+\alpha$. Let then $f=T+\alpha\in F[X]$. Then we have

[TABLE]

(using the interpretation of $\dim_{F}(I_{f})$ as the number of roots of unity where $f$ does not vanish), and consequently

[TABLE]

In particular, if $r=\operatorname{ord}_{p}(q)<\frac{p}{q}$, we obtain a counter example to the uncertainty principle for the field $E=\mathbb{F}_{q^{r}}$.

There exist infinitely many pairs of primes with this property. For instance, take $q=2$ and let $p$ be a prime such that the Legendre symbol $(\tfrac{2}{p})$ is equal to $1$. Then $q=2$ is a square modulo $p$, which implies that $2^{(p-1)/2}\equiv 1\bmod p$, hence that the order of $2$ modulo $p$ is $\leq(p-1)/2<p/2=p/q$.

More generally, fix the prime $q$ and take any prime $\ell>q$. By Lemma 2.7, if $p$ is any prime that is totally split in the Galois extension $K_{\ell}=\mathbb{Q}(e^{2i\pi/\ell},\sqrt[\ell]{q})$, we have $\operatorname{ord}_{p}(2)\leq(p-1)/\ell<p/q$. It is a well-known consequence of the Chebotarev density theorem that there are infinitely such primes.

In anticipation of the next section, we note however that, for any pair $q<p$ with $r<p/q$, it still remains true that

[TABLE]

or in other words, the uncertainty principle for $f$ does not fail drastically.

5. The weak uncertainty principle

5.1. Statement

The uncertainty principle in its current version over $\mathbb{C}$ states that for each prime $p$, we have $\mu_{\mathbb{C}}(p)>p$. We have seen that this inequality does not always hold if $\mathbb{C}$ is replaced by any field. Because of the link with good cyclic codes, we introduce a weaker version:

Definition 5.1** (Weak uncertainty principle).**

Let $\delta$ be a real number such that $0<\delta\leq 1$. We say that a field $F$ satisfies the $\delta$-uncertainty principle for a prime $p$ if

[TABLE]

This variant of the uncertainty principle is weaker than the one in the previous section in two respects: the lower bound for $\mu_{F,p}$ is relaxed, and it is stated with respect to an individual prime $p$, and not all of them.

Example 5.2**.**

We first present some finite fields that satisfy the weak uncertainty principle for certain primes. Let $\ell$ be a prime number, and let $P$ be an infinite set of primes such that $\ell$ is a primitive root in $\mathbb{F}_{p}^{*}$ for all $p\in P$. As we have already mentioned, Artin’s Conjecture asserts that such a set $P$ exists for any prime $\ell$, and Hooley [H] confirmed this under a suitable form of the Generalized Riemann Hypothesis. By Proposition 4.3, we have $\mu_{\mathbb{F}_{\ell}}(p)>p$, for any $p\in P$, and hence the weak uncertainly principle is satisfied by the field $\mathbb{F}_{\ell}$ for any prime in $P$.

This example does not however lead to good cyclic codes. Indeed, if we consider proper ideals $I_{p}\subset\mathbb{F}_{\ell}[\mathbb{Z}/p\mathbb{Z}]=\mathbb{F}_{\ell}[X]/(X^{p}-1)$ for $p\in P$, the fact that $\ell$ is a primitive root modulo $p$ means that $I_{p}$ is generated either by $X-1$ or by $(X^{p}-1)/(X-1)$. In the first case, we have $\dim I_{p}=p-1$, but the element $X-1$ has weight $2$, so that the distance of the code $I_{p}$ is $2$. In the second case, we have $\dim I_{p}=1$. In either case, the codes corresponding to $I_{p}$ are not good as $p\to+\infty$ in $P$ since one of the inequalities in (1.1) fails.

This example motivates our last variant of the uncertainty principle.

Definition 5.3** (Weak uncertainty principle, 2).**

Let $\delta$ and $\epsilon$ be real numbers such that $0<\delta\leq 1$ and $0<\epsilon<\delta$. We say that a field $F$ of size $\ell$ satisfies the $(\epsilon,\delta)$-uncertainty principle if there exists an infinite set of primes $P$ such that, for all primes $p\in P$, the two following conditions holds:

1. (1)

We have $\mu_{F,p}>\delta p$, 2. (2)

We have $\operatorname{ord}_{p}(\ell)<\epsilon p$.

The existence of finite fields $F$ which satisfy such an uncertainty principle implies the existence of good cyclic codes over $F$:

Theorem 5.4**.**

Let $F=\mathbb{F}_{\ell}$ be a finite field prime order $\ell$. Assume there exist real numbers $0<\epsilon<\delta<1$ such that $F$ satisfies the $(\epsilon,\delta)$-uncertainty principle. Then there exists an infinite family of good cyclic codes over the field $F$.

Proof.

For each prime $p\in P$, let $I_{p}\subset F[X]/(X^{p}-1)$ be a non-zero ideal such that

[TABLE]

Such an element exists because $r=\operatorname{ord}_{p}(\ell)<\epsilon p$ by definition, and $R=F[X]/(X^{p}-1)$ is a sum of ideals of dimension $r$ each, plus a one dimensional ideal, see Proposition 2.6 (3).

For every element $h\in I_{p}$, we have $I_{h}\subset I_{p}$ and hence $\dim(I_{h})\leq\dim(I_{p})$. From the weak uncertainty inequality that we assume, we get

[TABLE]

The cyclic code $I_{p}$ has length $p$; the last computation shows that its distance is $\geq(\delta-\epsilon)p$, and its dimension is $\geq\epsilon p/2$. Hence by definition (see (1.1)), the sequence $(I_{p})_{p\in P}$ is an infinite sequence of good cyclic codes over $F$.

∎

Generally speaking, condition (1) in Definition 5.3 ensures that we can find ideals with “large” distance, while condition (2) is used to show the existence of such ideals with “large” dimension.

Remark 5.5**.**

Our proof shows that any choice of ideal $I_{p}$, such that $\frac{\epsilon}{2}p\leq\dim(I_{p})<\epsilon p$ will give a good code. There are many possibilities for such ideals. This suggests that a randomized process might be used to prove existence of cyclic good codes even under a weaker uncertainty principle.

5.2. A uniform weak uncertainty principle does not hold

It is only natural to ask (and maybe hope) that a uniform weak uncertainty principle, uniform with respect to $\delta$, should hold for all finite fields, or in other words, to ask whether there exists $\delta>0$ such that $\mu_{F,p}>\delta p$ for any finite field $F$ and any prime $p$.

We will show – following an argument of Eli Ben-Sasson – that, assuming the existence of infinitely many Mersenne primes, this is not the case.

Proposition 5.6** (No uniform weak uncertainty principle).**

*Assume that there exist infinitely many Mersenne primes. Then, for any $\delta>0$, there exists a finite field $F$ and a prime number $p$ such that $\mu_{F,p}\leq\delta p$. *

For the proof, we will use the following result of Ore [O]:

Lemma 5.7** (Ore).**

Let $q$ be a prime number and $n\geq 1$. Let $F=\mathbb{F}_{q^{n}}$, and view $F$ as an $\mathbb{F}_{q}$-vector space of dimension $n$. For every integer $k\leq n$ and every $\mathbb{F}_{q}$-affine subspace $A\subset F$ of dimension $k$, the polynomial

[TABLE]

satisfies

[TABLE]

where $\alpha$ and $\alpha_{i}$ are elements of $F$. In particular, we have $\operatorname{wt}(f_{A})\leq k+2$.

Proof.

It is easy to see that it suffices to consider the case where $A$ is a vector subspace of dimension $k$. Then $f_{A}$ is a separable polynomial whose roots form an additive subgroup of $F$. This implies that $f_{A}$ is an additive polynomial (see [G, Th. 1.2.1]), which is necessarily of the desired form (with $\alpha=0$ in that case) by [G, Prop. 1.1.5]. ∎

Remark 5.8**.**

In general, if $K$ is any field, an additive polynomial $f\in K[X]$ is a polynomial such that $f(x+y)=f(x)+f(y)$ for any $x$ and $y$ in $K$. If $K$ has characteristic zero, it is easy to check that $f$ is necessarily of the form $f=aX$ for some $a\in K$, but this is not so in characteristic $p>0$, since any monomial $X^{p^{i}}$ is then an additive polynomial. The result we used is that any additive polynomial is a linear combination of these monomials.

Proof of Proposition 5.6.

Let $q=2$ and let $p=2^{n}-1$ be a Mersenne prime, so that $n=\operatorname{ord}_{p}(2)$. Let $F=\mathbb{F}_{2^{n}}$. Then the non-zero elements of $F$ are precisely the $p$-th roots of unity.

We view $F$ as an $n$-dimensional vector space over $\mathbb{F}_{2}$, and fix a basis $e_{1}$, …, $e_{n}$. Let $k$ be an integer parameter such that $1\leq k<n$.

There exist disjoint affine subspaces $A_{1}$, …, $A_{k}$ in $F$, none of which contains [math], with $\dim(A_{i})=n-i$ (for instance, we could take $A_{i}$ to be the subspace defined by the equations

[TABLE]

where $(x_{1},\ldots,x_{n})$ are the coordinates of an element $x$ of $F$ with respect to the chosen basis $(e_{1},\ldots,e_{n})$).

The disjoint union of the subspaces $A_{i}$ has cardinality

[TABLE]

Thus if we denote by $f_{i}$ the polynomial associated to $A_{i}$ as in Lemma 5.7, and put

[TABLE]

then we have

[TABLE]

since $1\leq k<n$ and

[TABLE]

Since $\gcd(f,X^{p}-1)=f$, we have

[TABLE]

Let $\delta>0$ be any given real number. Take some integer $k\geq 1$ such that $\frac{1}{2^{k}}\leq\frac{\delta}{2}$. By the assumption that there exist infinitely many Mersenne primes, we can find a prime $p=2^{n}-1$ for which $n>k$ and

[TABLE]

Then using the polynomial $f$ obtained as above for these parameters $p=2^{n}-1$ and $k$, we get

[TABLE]

and therefore $\mu_{F,p}\leq\delta p$. ∎

It is important to notice that this counter-example does not show that $\mathbb{F}_{2}$ does not satisfy the $\delta$-uncertainty principle for the prime $p$, since the polynomials $f_{i}$ and $f$ do not usually belong to $\mathbb{F}_{2}[X]$. Furthermore, as the underlying field depends on the primes $p$, this counter example is not really relevant to our search of families of cyclic good codes, since in such a family we need to work with a fixed underlying field while in the last example, the size of $F$ grows to infinity.

6. Why good cyclic codes should exist

6.1. Preliminaries

In this section, we describe some heuristic arguments that all point in the direction of the existence of families of good cyclic codes, and of the weak uncertainty principle according to Definition 5.3.

In both arguments, the main unproved claim is that for a polynomial of degree $<p$, the property of being “sparse” (i.e., of having small weight $\operatorname{wt}(f)$) and of vanishing on many roots of unity should be roughly independent. The following result is then relevant.

Lemma 6.1**.**

Let $\delta$ be a fixed real number with $0<\delta<1/2$. Let $S_{\delta}$ be the set of polynomials $f$ in $\mathbb{F}_{2}[X]/(X^{p}-1)$ with $\operatorname{wt}(f)\leq\delta p$. Then we have

[TABLE]

where $H^{\prime}(\delta)=H(\delta)/\log(2)$ and

[TABLE]

is the entropy for Bernoulli random variables.

Sketch of proof.

We have

[TABLE]

which the Stirling formula reveals to be of size

[TABLE]

as claimed. ∎

We also recall some fairly classical results on primes where $2$ has relatively small multiplicative order.

Lemma 6.2**.**

(1)* For any $\epsilon$ with $0<\epsilon<1$, there exist infinitely many primes $p$ such that $\operatorname{ord}_{p}(2)<\epsilon\cdot p$.*

(2)* Assume the Generalized Riemann Hypothesis for Dedekind zeta functions of number fields. For any $\epsilon>0$, there exist infinitely many primes $p$ such that $\operatorname{ord}_{p}(2)<p^{3/4+\epsilon}$.*

Proof.

In both cases, we use the criterion of Lemma 2.7: if $\ell$ is an odd prime and if $p$ is an odd prime distinct from $\ell$ such that $p$ is totally split in the field $K_{\ell}=\mathbb{Q}(e^{2i\pi/\ell},\sqrt[\ell]{2})$, then $p\equiv 1\pmod{\ell}$ and the order of $2$ modulo $p$ divides $(p-1)/\ell$, hence is $<p/\ell$.

Hence, taking $\ell$ to be any prime such that $\ell>1/\epsilon$, the first statement follows from the existence of infinitely many primes totally split in $K_{\ell}$ (this is an easy consequence of the Chebotarev Density Theorem, see for instance [N, Th. 13.4]).

For the second, we use the explicit form of the Chebotarev Density Theorem, following Serre’s presentation of the results of Lagarias and Odlyzko: for any odd prime $\ell$ and any $X\geq 2$, the number $\pi_{\ell}(X)$ of primes $\leq X$ which are totally split in $K_{\ell}$ satisfies

[TABLE]

where the implied constant is absolute, under the assumption that Dedekind zeta functions satisfy the Riemann Hypothesis. Precisely, this follows from [S, Th. 4], applied with $E=K_{\ell}$, $K=\mathbb{Q}$ and $C$ the trivial conjugacy class of the identity element; then $n_{E}=[K_{\ell}:\mathbb{Q}]$ and the discriminant $d_{E}$ is estimated using the bound [S, (20)].

In particular, since the integral is of size $X/(\log X)$ and $[K_{\ell}:\mathbb{Q}]\leq\ell^{2}$, this result shows that if $\epsilon>0$ is fixed and $\ell$ is any prime large enough, there exists a prime $p$ totally split in $K_{\ell}$ with $p\leq\ell^{4+\epsilon}$. Such a prime $p$ satisfies

[TABLE]

and the result follows. ∎

The interest of these statements is that if the order $r$ of $2$ modulo $p$ is “small” compared with $p$, then by the discussion following Proposition 2.6, the ring $R=\mathbb{F}_{2}[X]/(X^{p}-1)$ contains many ideals. In particular, if $r=p^{3/4+\epsilon}$ and $\eta$ with $0<\eta<1$ is fixed, and if we look for ideals of dimension $ir\approx\eta p$, then for such primes we have approximately $\binom{s}{i}$ ideals of dimension $\eta p$, where (see Proposition 2.6), we have $s=(p-1)/r$ and $i=\eta p/r\sim\eta s$. By Stirling’s formula, as in the Lemma 6.1, this numbers grows exponentially with $s$.

6.2. Picking ideals at random

Fix some real number with $0<\eta<1$. Let $p$ be a prime such that there exists an ideal $I$ in $R=\mathbb{F}_{2}[X]/(X^{p}-1)$ with $\dim(I)\sim\eta p$.

Let $\delta>0$ be another parameter. Assuming that the probability for an element of $I_{p}$ to be in the set $S_{\delta}$ of Lemma 6.1 is approximately the same as the probability for a general element of $R$, the expected cardinality of the intersection $S_{\delta}\cap I$ should be about

[TABLE]

by Lemma 6.1. If $\eta$ and $\delta$ are chosen so that

[TABLE]

this expectation is $<1$. So, as in the Borel-Cantelli lemma, if we select an ideal $I_{p}$ of this approximate dimension for all primes where this is possible (an infinite set, by Lemma 6.2 and Proposition 2.6), we may expect that only finitely many $p$ will have the property that $I_{p}$ intersects $S_{\delta}$. Since $H^{\prime}(\delta)\to 0$ as $\delta\to 0$, a suitable choice of $\delta$ exists for any fixed $\eta$.

Moreover, under the Generalized Riemann Hypothesis, picking the primes $p$ as given by Lemma 6.2 (2), the number of options for $I_{p}$ grows exponentially as a function of $s=p/\operatorname{ord}_{p}(2)\approx p^{1/4-\epsilon}$, and we need to succeed only with a single one of them to obtain a good cyclic code with rate $\eta$.

6.3. The weak uncertainty principle should hold

Here we give a heuristic argument, suggested by B. Poonen, as to why the weak uncertainty principle of Definition 5.3 should hold for the field $\mathbb{F}_{2}$ for an infinite sequence of primes. This is a variant of the previous argument.

First, the Generalized Riemann Hypothesis implies that there are infinitely many primes such that $\operatorname{ord}_{p}(2)=\frac{p-1}{2}$ (this is a simple variant of the argument of Hooley [H] for primitive roots, where we count primes that are split in the quadratic field $\mathbb{Q}(\sqrt{2})$, and not split in any field $\mathbb{Q}(e^{2i\pi/\ell},\sqrt[\ell]{2})$ for $\ell\geq 3$ prime, see Lemma 2.7 and [Mo]).

We consider such primes and explain that all but finitely many should satisfy Definition 5.3 with $\epsilon=1/2$ and $\delta=3/5$. Indeed, the condition $\operatorname{ord}_{p}(2)<\epsilon p$ holds by construction. Suppose $\mu_{F,p}\leq\delta p$. Then there exists a non-zero $f\in\mathbb{F}_{2}[X]$ of degree $<p$ such that

[TABLE]

Since $\operatorname{ord}_{p}(2)=(p-1)/2$, the polynomial $(X^{p}-1)/(X-1)$ has exactly two irreducible factors of degree $(p-1)/2$. So the gcd of $f$ and $X^{p}-1$ is of degree $1$, $(p-1)/2$ or $p-1$. In the first case, the inequality (6.1) is clearly false. In the third case, we have $f=(X^{p}-1)/(X-1)$, with $\operatorname{wt}(f)=p$, and again (6.1) is false. So $f$ must be divisible by exactly one of the two factors of degree $(p-1)/2$, say $f_{1}$, and then we must have $\operatorname{wt}(f)\leq p/10+1/2$ for (6.1) to hold.

Now comes the heuristic argument, where we will assume that the property of being divisible by $f_{1}$ and of having support of size $\leq p/10$ are “independent”: the number of polynomials $f$ of degree $<p$ divisible by $f_{1}$ is about $2^{p/2}$, and on the other hand, the number of polynomials $f$ of degree $<p$ with $\operatorname{wt}(f)<p/10$ is $2^{pH^{\prime}(1/10)+o(p)}$ by Lemma 6.1. Since

[TABLE]

we may hope that the expected number of polynomials in the intersection is

[TABLE]

and since the sum of the series $\sum 2^{-3p/100}$ is finite, this suggests (by analogy with the Borel-Cantelli lemma) that the set of primes where the intersection is non-empty is finite.

F. Voloch has pointed out that one must be careful with this heuristic. Indeed, let $C_{p}$, for $p$ odd, be the quadratic residue code of dimension $(p-1)/2$, namely the cyclic code corresponding to the principal ideal generated by the polynomial

[TABLE]

If the last step is taken literally, the previous argument suggests that the family of the cyclic codes $C_{p}$, parameterized by primes $p$ such that $\mathrm{ord}_{p}(2)=(p-1)/2$, is good. However, assuming GRH, Voloch’s results [V] imply that this is not the case.

More precisely, Voloch shows, under the Generalized Riemann Hypothesis, that there exist an infinite sequence of primes $p$ for which the distance of the code $C_{p}$ is $\ll p(\log p)^{-1}$ (he obtains an unconditonal bound of size $\ll p(\log\log p)^{-1}$). Although the primes that he constructs in [V] do not necessarily satisfy the condition $\mathrm{ord}_{p}(2)=(p-1)/2$ that we wish to impose, we will now show that the two can be combined (as was suggested to us by Voloch).

Indeed, Voloch defines a sequence of Galois extensions $L_{\ell}/\mathbb{Q}$ of degree about $(\ell-1)2^{\ell}$, for $\ell$ a prime. He shows that if $p$ is totally split in $L_{\ell}$, then the distance of $C_{p}$ is $\leq(p-1)/(2\ell)$ (for this purpose, he uses a formula of Helleseth). It turns out that the splitting restrictions in $L_{\ell}$ are compatible with those involved in constructing primes with $\mathrm{ord}_{p}(2)=(p-1)/2$. Under the Generalized Riemann Hypothesis, one gets by following Hooley’s method (see, e.g., [Mo, §5]) that for a given odd prime $\ell$ and for $X\geq 2$, there are roughly

[TABLE]

primes $p\leq X$ satisfying all the desired combined splitting conditions. Since the degree of $L_{\ell}$ over $\mathbb{Q}$ is about $\ell 2^{\ell}$, we can find a prime $p$ of size about $\exp(\exp(\ell))$ that satisfies the desired conditions. This provides an infinite family of codes $C_{p}$ with distance $\ll p/(\log\log p)$, under the Generalized Riemann Hypothesis.

Although this discussion shows that the heuristic argument cannot be literally correct, the optimist might still hope that the events which we consider are sufficiently independent to still lead to infinitely many primes where the weak uncertainty principle holds. It is maybe a positive sign that the primes given by Voloch’s argument are rather sparse, and even then, only a very slow decay of their distance is proved.

Appendix

Chebotarev’s Theorem

A well-known (but not the best-known!) result of Chebotarev [C] states the following:

Theorem 6.3** (Chebotarev).**

Let $p$ be a prime and $\xi=e^{\frac{2\pi i}{p}}\in\mathbb{C}$. Let $V$ be the Vandermonde matrix $V=(\xi^{ij})_{i,j=0}^{p-1}\in M_{p}(\mathbb{C})$. Then each minor of the matrix $V$ is invertible, i.e., we have $\det(V|_{A\times B})\neq 0$ for any $A,B\subset\{0,\ldots,p-1\}$, $|A|=|B|$, where $V|_{A\times B}$ denotes the minor of $V$ with rows in $A$ and columns in $B$.

Let $R=\mathbb{C}[X]/(X^{p}-1)$. Then $R$ is a vector space over $\mathbb{C}$ with basis the images of the monomials $e_{i}=X^{i}$ for $0\leq i\leq p-1$.

(A multiple of) the Fourier transform on $\mathbb{Z}/p\mathbb{Z}$ can be interpreted as the linear map $\mathcal{F}\colon f\mapsto\widehat{f}$ from $R$ to $R$ such that

[TABLE]

It is elementary that the matrix representing this linear map is $V^{\prime}=(\xi^{-ij})_{i,j=0}^{p-1}\in M_{p}(\mathbb{C})$. Then each minor of the matrix $V$ has a non-zero determinant if and only if the same property holds for the matrix $V^{\prime}$, so we may replace $V$ by $V^{\prime}$ in proving Chebotarev’s Theorem.

We now show that Theorem 6.3 is equivalent to the uncertainty principle over $\mathbb{C}$. For a direct simple proof of Chebotarev’s Theorem, see the note [F] of Frenkel.

Proposition 6.4**.**

Chebotarev’s Theorem 6.3 is equivalent to the uncertainty principle for $\mathbb{Z}/p\mathbb{Z}$ over $\mathbb{C}$, i.e., to Theorem 3.2.

Proof.

For each $A\subset\{0,\ldots,p-1\}$, we denote by $\ell^{2}(A)$ the space of elements of $R$ which have zero coefficients for the basis vectors $e_{i}$ for $i\notin A$, i.e., polynomials $f$ with support contained in $A$. For an element

[TABLE]

we denote by $f|_{A}$ the element

[TABLE]

of $\ell^{2}(A)$.

For any two subsets $A$ and $B$ of $\{0,\ldots,p-1\}$ with the same cardinality, the linear map $T_{A,B}\colon\ell^{2}(A)\to\ell^{2}(B)$ obtained by restricting the Fourier transform (i.e., $T_{A,B}(f)=\widehat{f}|_{B}$ for $f\in\ell^{2}(A)$) is represented by the matrix $V^{\prime}_{A\times B}$ with respect to the bases $(e_{i})_{i\in A}$ and $(e_{i})_{i\in B}$.

(Theorem 6.3 $\Rightarrow$ Theorem 3.2) Assume for contradiction that there exists a non-zero element

[TABLE]

such that $|\operatorname{supp}(f)|+|\operatorname{supp}(\widehat{f})|\leq p$. Let $A=\operatorname{supp}(f)$. Since $|\operatorname{supp}(\widehat{f})|\leq p-|A|$, the complement of $\operatorname{supp}(\widehat{f})$ has cardinality $\geq|A|$. We can therefore find a subset $B$ of the complement of $\operatorname{supp}(\widehat{f})$ such that $|B|=|A|$. Let $T=T_{A,B}:\ell^{2}(A)\rightarrow\ell^{2}(B)$. We then have $T(f)=\widehat{f}|_{B}=0$ since $B$ is in the complement of the support of $\widehat{f}$, but $f$ is non-zero in $\ell^{2}(A)$. Hence $T$ is not invertible. Hence, by the previous remark, the matrix $V^{\prime}_{A\times B}$ has determinant zero, which contradicts Chebotarev’s Theorem.

(Theorem 6.3 $\Leftarrow$ Theorem 3.2) Now assume that there exist subsets $A,B\subset\{0,\ldots,p-1\}$ with $|A|=|B|$ and $\det(V^{\prime}|_{A\times B})=0$. This means that the linear map $T=T_{A,B}:l^{2}(A)\rightarrow l^{2}(B)$ is not invertible. In particular, $T$ is not injective. Let $f\not=0$ be an element of $\ell^{2}(A)$ such that $0=T(f)=\widehat{f}|_{B}$. Then $\operatorname{supp}(f)\subset A$ and $B$ is contained in the complement of the support of $\widehat{f}$. Hence

[TABLE]

which contradicts the uncertainty principle. ∎

In this argument, we may replace $\mathbb{C}$ with any other field $F$ containing a $p$-primitive root of unity $\xi$. So for any prime $p$ and for any field $F$ containing a $p$-primitive root of unity $\xi$, Theorem 6.3 with respect to the prime $p$ (i.e. the claim that each minor of the $p\times p$ Vandermonde matrix $(\xi^{ij})_{i,j}$ is invertible) is equivalent to the uncertainty principle for the field $F$ with respect to $p$, i.e., to the claim that $\mu_{F,p}>p$.