1. Introduction
Let be positive integers. A locally recoverable code (LRC) having parameters is an subspace of of dimension such that, if one deletes one component of any , this can be recovered by accessing at most other components of . LRC codes are of great interest in the context of distributed and cloud storage systems. One of the most interesting constructions of LRC is due to Tamo and Barg [tamo2014family] and is realised via constructing polynomials of degree which are constants on subsets of of cardinality . These polynomials are called good polynomials. Construction of good polynomials are also provided in [liu2018new]. All these constructions are essentially based on algebraic properties of the base field .
In this paper we fit the theory of good polynomials in a Galois theoretical context, showing that finding polynomials that are good can be reduced to solve a Galois theory problem. Moreover, existing constructions of good polynomials fit completely in our context and can be derived by our main theorems in Section 3.
In addition, using the same method we provide existential results in Subsection 3.1 and explicit new constructions in Subsection 3.2. In Section 4
we put the method in practice constructing good polynomials for many different base fields. All the results are based on estimates provided by the Chebotarev Density Theorem for global function fields, which are also supported by the tables in Section
4.We explain now in a nutshell the key idea of our constructions. We are interested in polynomials of degree which are constants on disjoint sets (i.e. , for some ) each of cardinality , for , for some . One of the tasks is to maximise , as this will allow to build many LRC codes, as it is explained in Theorem 2.5 (of course, it is trivial to get , as it is enough to take a totally split polynomial). The fundamental observation is that is exactly the number of totally split places of degree of the global function field extension , where is trascendental over and is a zero of over the algebraic closure of . Now, a place is totally split in if and only if it is totally split in , where is the Galois closure of the extension . Let be the Galois group of . By the Chebotarev Density Theorem (and under some additional technical requirements), the number of totally split places is roughly of the size of . Therefore, the entire task of finding good polynomials relies on finding polynomials with minimal Galois group .
The strenght of our method relies on the fact that we do not need to base our constructions on algebraic properties of the base field as in [liu2018new, tamo2014family] but we extract good polynomials with a density argument. Let us now show with a toy example this fact. Suppose that we want to construct via good polynomials a LCR code over an alphabet of size , with using one of the constructions in [liu2018new]. One would need a degree polynomial that is totally split at at least some place . But then this cannot be a composition, as its degree is prime and cannot be a linearised polynomial, as its degree is incompatible with the characteristic. Also, it cannot be a power because is a bijection by the congruence class of modulo , so the constructions in [liu2018new] do not apply. Of course, one could get an LRC code by extending the base field to (instead of ) and using the good polynomial . Our approach does not need field extensions: for example Theorem 3.11 always ensures that the existence (and constructibility) of a good polynomial of degree with predictable without increasing the field size. If and are fields contained in a larger field , we denote by their compositum, i.e. the smallest subfield of containing and .
Notation
For us, a LRC code is a subspace of of dimension with locality , i.e. if a component of a codeword is lost, then it can be recovered by looking at most at other components. Let be fields. We denote by the polynomial ring in the variable over the field . A field extension will be denoted by (and not , in order not to overlap with other notation) and its degree by , i.e. the dimension of as a
vector space. For any
that is not a square, we denote by any zero of over the algebraic closure of . Let be a prime number, a positive integer, and be the finite field of order . Let be trascendental over and be the rational function field in the variable over the base field . In this paper we use the notation and terminology of [stichtenoth2009algebraic], which we briefly recall here. A global function field over is a finite dimensional extension of . For a place of , we denote by its valuation ring and say that has degree if . We will only deal with global function fields, so the term global will mostly be understood.For a function field over , we denote by (resp. ) the set of places (resp. places of degree ) of . Let be an extension of function fields. Let be a place of and be a place of . We say that lies above if . Moreover we denote by (resp. ) the ramification index (resp. the relative degree) of the extension of places . We say that is a separable polynomial if , in such a way that is a separable irreducible polynomial over . We denote by the splitting field of over , i.e. the Galois closure of the extension , where is any of the roots of over the algebraic closure of . We denote by the field of constants of , i.e.
Notice that in principle might be non trivial (an example is for and : in this case and ). Recall that . Let be the monodromy group of , i.e. the Galois group of the the field extension . When we refer to the genus of we consider as a function field over its field of constants . In all interesting cases we will have so this distinction will not affect us. For an element and a place we say that is a Frobenius for if there exists a place lying above such that and the map acts as . In particular, we say that the identity is a Frobenius for if .
In the rest of the paper we will identify the places of degree of with . For a finite set , we denote its cardinality by . Let us denote the symmetric group of degree by and the alternating group by . Let us denote the multiplicative group by . We say that a polynomial is totally split if it factors as a product of distinct linear factors.
2. Locally Recoverable Codes and Good Polynomials
Let us start with the fundamental definition.
Definition 2.1.
Let be a polynomial of degree and let be a positive integer. Then is said to be good if

has degree ,

there exist distinct subsets of such that

for any , for some , i.e. is constant on ,

,

for any .

We say that the family is a splitting covering for . We say that a polynomial is good if it has degree and it is good for some . For simplicity of notation, we allow to be negative or zero, in which case an good polynomial is simply a polynomial of degree . A polynomial that is not good is a polynomial such that there is no such that splits completely in distinct linear factors.
Remark 2.2.
Observe that if a polynomial of degree is good, then is at most .
Remark 2.3.
Notice that an good polynomial is also good for any , as one can simply drop some of the ’s.
Let us recall the definition of optimal LRC codes [tamo2014family]
Definition 2.4.
A LRC code is said to be optimal if the minimum distance of satisfies
In fact it can be proven that is the maximum distance achievable by any LRC code [papailiopoulos2014locally].
The following result is [tamo2014family, Construction 1]. We write it in the format of [liu2018new, Theorem 4], which is more convenient for our purposes.
Theorem 2.5.
Let be a positive integer and be an good polynomial over for the set . Let , and . For , let
Define
Then is an optimal LRC code over .
The following is the key observation.
Remark 2.6.
Let , where is any root of in . It is easy to see that each of the ’s corresponds to a totally split place of degree of the extension : if for some , then the place corresponding to factors as a product of exactly places of , which themselves correspond to the elements of .
Clearly, the correspondence between the ’s and the totally split places of is injective and simply given by . In addition, the maximum for which is good is the number of totally split places in of the extension . As Theorem 2.5 shows, a large is desirable as for fixed locality it allows constructions of optimal codes with parameters .
Remark 2.7.
Notice that it is obvious to construct an good polynomial as it is enough to take a totally split polynomial. That allows to construct only one LRC code with parameters . This is the reason why in this paper we include both and in the notion of “good” polynomial.
3. Galois Theory over Global Function Fields and Good Polynomials
We now briefly explain the essence of the method. We start with a polynomial and we are interested in the number of ’s in such that splits completely, by Remark 2.6. Let be trascendental over and let us consider the extension where is a root of over the algebraic closure of . The splitting of the places of degree of the extension correspond to the factorization shapes of , when varies in . Unfortunately, the extension is not always Galois, but if we take the Galois closure of such extension, we can still extract information about the splitting of the places in by looking at the disjoint cycle decomposition of the elements of the Galois group (this is a classical fact, but see for example [ferraguti2018complete, Lemma 1] or [micheli2017selection, Theorem 6]): as we are interested in the totally split places, we take the identity as element (which has all fixed points). As long as the field of constants of is simply , Chebotarev Density Theorem for function fields applied on the identity (see for example [kosters2017short]) ensures that the number of totally split places can be precisely estimated by [kosters2017short, Corollary 1].
The following proposition summarises all the results we need from algebraic number theory in a compact way. We include the proof for completeness.
Proposition 3.1.
Let be a separable polynomial. Let be the genus of and let be a root of in .

If the extension field has a totally split place of degree , then .

Suppose that . Let be the set of such that factors in distinct linear factors over . Then
where is the set of ramified places of degree of the extension .

Let be the smallest integer such that
If , and , then has a totally split place.
Proof.
Let us prove (i). Since is the Galois closure of , by [stichtenoth2009algebraic, Lemma 3.9.5.] is totally split also in . Let be a place lying above the totally split place . Since is totally split and of degree we have that and then . By [stichtenoth2009algebraic, Proposition 1.1.5, (c)] we have that contains the field of constants of , and in turn cannot be a proper extension of .
Let us now prove (ii). Let be a place of . Since is a Galois extension of all places of lying above have the same relative degree and ramification index by [stichtenoth2009algebraic, Corollary 3.7.2].
Claim 1. A place in is totally split in if and only if it is unramified and the identity is a Frobenius for .
Proof of claim 1.
A place is totally split if and only if any place lying over is unramified and , which happens if and only if is unramified and the identity of acts as a Frobenius for the (trivial) field extension . ∎
Claim 2. Let . The polynomial splits into distinct linear factors if and only if the place corresponding to in is totally split in .
Proof of claim 2.
Recall that is a root of . First observe that the splitting of degree places in the extension correspond exactly to the factorization of , for . Since is the Galois closure of the extension , then [stichtenoth2009algebraic, Lemma 3.9.5.] ensures that is also totally split in . Viceversa, since ramification and relative degrees are multiplicative in intermediate extensions, it is easy to see that if is totally split in the extension , then it is also totally split in . ∎
Using the claims above and observing that the place at infinity of is ramified, we reduced the problem to finding all places that are totally split in .
We want to use [kosters2017short, Corollary 1]. Since in our case the field of constants extension is trivial. In the notation of Koster’s corollary we have that , , and . Moreover, we are interested in , therefore , where if the identity is a Froebenius at (i.e. splits into factors of relative degree and multiplicity ), and otherwise. Then we have
Splitting the sum for ramified and unramified places we get
Observing now that , the place at infinity is ramified, and
the final claim follows directly.
The statement in (iii) is immediate by observing that the condition on ensures the existence of a totally split place by point (ii). ∎
Point (ii) of the proposition above essentially states a very classical fact from algebraic number theory: since the number of ramified places is bounded by an absolute constant (depending only on the degree of ) the set of totally split places of a Galois extension of global fields has density where is the Galois group of the extension field. When the global fields are actually global function fields (i.e. finite extensions of ) the number of degree one totally split places can be estimated as in [kosters2017short, Corollary 1], leading to the estimate in (ii). The estimate is essentially optimal, as the Riemann Hypothesis for curves over finite fields is proved and is equivalent to the HasseWeil bound.
We notice now how and can be explicitely bounded by a constant depending on the degree of and independent of .
Remark 3.2.
Proposition 3.1, point (ii) is in the format of an estimate, but whenever is zero, it can be used to obtain an exact formula, if in the proof one works out exactly what happens at the ramified places for the quatities .
Proposition 3.3.
Let be a separable polynomial, , and be the genus of the splitting field of over . Let (resp. ) be the set of ramified places of (resp. ramified places of degree ) in the field extension . Then

.

Suppose that and . Then .
Proof.
In (i), the first inequality is obvious by inclusion of sets. Let be a root of in . The second inequality comes from the fact that a place of is ramified in the extension if and only if it is ramified in its Galois closure . Therefore, it is enough to look at the zeroes of the derivative of to find the ramified places at finite, which are at most . The place at infinity is always ramified.
To prove (ii) we want to use Hurwitz genus formula in [stichtenoth2009algebraic, Corollary 3.4.14.]. Since the characteristic is coprime with the degree of the extension and since (because is a Galois extension of ), we can use Dedekind Different Theorem in the tamely ramified case [stichtenoth2009algebraic, Theorem 3.5.1, (b)] obtaining
restricting the outer sum to ramified places and using the trivial estimate and the fact that we have that
Using now the fundamental equality [stichtenoth2009algebraic, Theorem 3.1.11] we get
Now, the ramified places at finite correspond to the evaluations of at the zeroes of . Since has degree and the place at infinity is ramified of degree , we have that . The final claim follows immediately. ∎
Remark 3.4.
The condition is purely technical and needed to obtain explicit estimates later on.
3.1. Existence of good polynomials
In this section we prove some existential results over base fields which are relatively large compared with the locality parameter .
Proposition 3.5.
Let be a positive integer and be a prime power. Then any separable polynomial of degree over such that is at least good, with at least
Moreover, if then is not a good polynomial.
Proof.
First observe that can be seen as a subgroup of the symmetric group , which forces . The statement follows immediately by applying point (ii) of Proposition 3.1 and bounding with point (i) of Proposition 3.3. If , then the statement (i) of Proposition 3.1 applies, so cannot have a totally split place and therefore it cannot be good for any positive integer .
∎
Remark 3.6.
The condition is generic, let us briefly sketch the reason here. Let by the primitive element theorem, with satisfying a degree polynomial over . We have if and only if is irreducible [stichtenoth2009algebraic, Corollary 3.6.8] and the condition of being irreducible for a bivariate polynomial over a finite field is generic.
Proposition 3.7.
Let be a set of size . Let be a positive integer such that and . Then is good, with at least
Proof.
Clearly is separable because the characteristic does not divide degree of the polynomial. First using (i) of Proposition 3.1 we get that , as is a totally split place. Now using Proposition 3.5 we get that is at least good. Using now the bound on given by (ii) of Proposition 3.3 we get the wanted result. ∎
Remark 3.8.
The proposition above implies that forcing just one totally split place immediately gives the existence of many totally split places, which is an interesting fact. It is worth noticing here that the worst case scenario given in Proposition 3.7 is actually the generic case: if one fixes a random polynomial of degree in and considers , then most likely , in which case the estimate for the number of totally split places is given by . Table 2(b) shows some instances of this fact for degree .
3.2. Selection of very good polynomials
The method provides existential results and fits the existing literature on good polynomials in a Galois theoretical context, but also allows to produce new good polynomials, which is the most important application. We give emphasize here that we want “very good” polynomials, i.e. good polynomials such that is as large as possible (we already noticed in Remark 2.7 that building an good polynomial is a trivial taks if is not taken into account). Let us start with the two fundamental and well known constructions
Proposition 3.9.
The following are good polynomials

Let be a divisor of . The polynomial is good.

Let be an additive subgroup of . The polynomial is good.
If one combines the construction above via composition, one can get new good polynomials as described in [tamo2014family, Theorem 3.3] and [liu2018new, Section A,B].
Remark 3.10.
All the good polynomials above have in common that and , so thanks to Remarks 3.2 and 2.6, they fit completely in the easy case (the genus zero condition) of our context. In particular, (which is a necessary condition for a polynomial to be good) exactly when or the linearised polynomial splits completely over .
Also, the splitting field of their composition inherits nice properties so the results in [tamo2014family, Theorem 3.3] and [liu2018new, Section A,B] can also be derived from our framework.
For the sake of explanation of our method, let us fit for example the case of in our context. For simplicity, let us assume . First, observe that such example of good polynomial exists only when , which is exactly the condition needed to have : in fact, if then contains a primitive th root of unity and therefore we have that as it is enough to add just one root of to to get all the other roots. Now, as is just another name for so the function field is still rational, so is zero. Trivially . In addition so we get
Since has to be an integer and , we obtain directly that . In this case the direct proof is actually easier than the one proposed here, on the other hand it only works thanks to the cyclic group structure of while our approach works in general, as it is based on a density argument.
The method. Constructing good polynomials of given degree using Proposition 3.1 is very simple: the number of ’s such that splits into distinct linear factors can be estimated with (as long as , otherwise it is the empty set) up to an error term depending on the ramified places of degree , and . Informally, to construct an good polynomial the task becomes the following: find a transitive subgroup of such that is roughly of the size of and realise as for a family of ’s. Now sieve the family allowing only polynomials with minimal ramification. We give now simple applications of the method in the case with , in the case with , and with . We want to stress here that these constructions hold for almost all ’s.
Theorem 3.11.
Let be a prime power, and . Then

if is even then is good with at least

if , then is good with at least
Moreover, if then is good with at least
Proof.
Applying (i) of Proposition 3.1 gives that and therefore we can apply (ii). Clearly is at most , as it has to be a subgroup of the symmetric group. is constructed by simply adding two roots of , since the third one can always be obtained from the other two with field operations. Therefore ( can be obtained by evaluating at for example), and then by Riemann’s inequality [stichtenoth2009algebraic, Corollary 3.11.4] (the minimal polynomial of over has degree at most and viceversa). The ramified places at finite of are in correspondence with the zeroes of the derivative of . As usual, the place at infinity is always ramified.

For even , so that .

For , , which has either no roots ( and ) or one root ( and ).

For odd , we have , which has no roots exactly when is not a square.
Using the formula in (ii) of Proposition 3.1 we get the wanted results.
∎
Remark 3.12.
Notice that the construction is not exploiting the multiplicative nor the additive subgroups of . With this method we are able for example to write a polynomial of fixed degree (see for example the case of Theorem 3.11) in that is good at any , with being an explicit constant.
Suppose now that we want to construct a code over an alphabet of size , with . For some we would need an good polynomial . None of the constructions in [liu2018new] apply as we now explain. First of all, observe that cannot be a composition of a nontrivial linearised polynomial and a power function, as its degree is and is odd. If was a power function, then (up to multiplication by a scalar) , but then and so is never totally split for any . The following result provides a good polynomial with roughly of the size of .
Theorem 3.13.
Let be an odd prime power. Let and . Then and

if is not a square in , the polynomial is good with at least

if is a square, then is at least
Proof.
First we compute the splitting field of . It is clear that
Since , then we have that
By the tower law we have that . In order to show that it is enough to show that . This can happen only when is Galois over , and therefore . Consider the subfields and . and are distinct and satisfy . By contradiction, let us assume . One observes that, since then , and therefore can be written as a compositum of and , i.e. . But one can show directly that is not a square in , so does not contain all the roots of and therefore cannot be its splitting field.
We now compute and . For simplicity let us denote and . One can show that . Also, since can be obtained using only and elementary operations that do not involve . The elements and verify the equation , which is a conic so its genus is zero. Another way to see the fact that is to use the inequality in [stichtenoth2009algebraic, Proposition 3.11.5] on the equation . Moreover by [stichtenoth2009algebraic, Corollary 3.6.8], because is absolutely irreducible if .
Let us now compute . Since the ramified places of degree correspond to the zeroes in of the derivative of (the place at infinity is always ramified), it is easy to see that if is not a square in and otherwise, where is an element such that . A direct application of Proposition 3.1 gives now the wanted result.
∎
Let us finish with an example of a good polynomial. Again, let us explain why this is a new example of a good polynomial. Fix the size of the base field to be and assume that one wants to construct an LRC with locality . Suppose that and is not divisible by or . Then one would need a degree polynomial such that is totally split for many ’s in . But then, none of the constructions in Proposition 3.9 will work, nor compositions of those: in fact cannot be a composition (possibly trivial) of a linearised polynomial with a power function for degree reasons, and also cannot be power function because , and therefore is not a good polynomial over .
Theorem 3.14.
Let be an odd prime power such that and such that in not a square. Let . Then and is a good polynomial with at least
Proof.
First, we need to compute , the splitting field of . Observe that and that adding to allows the splitting
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