On Parameters of Subfield Subcodes of Extended Norm-Trace Codes

In this article we describe how to find the parameters of subfield subcodes of extended Norm--Trace codes. With a Gr\"obner basis of the ideal of the $\mathbb{F}_{q^r}$ rational points of the Norm--Trace curve one can determine the dimension of the subfield subcodes or the dimension of the trace code. We also find a BCH--like bound from the minimum distance of the original supercode.


Background and motivation
Extended Norm-Trace codes were introduced in [1]. Extended Norm-Trace codes generalize Norm-Trace codes and Hermitian codes. Our aim is to understand subfield subcodes of extended Norm-Trace codes. These codes include algebraic geometry codes from quasi-Hermitian curves and Norm-Trace curves. BCH codes and binary classical Goppa codes are subfield subcodes of RS codes and subfield subcodes of special algebraic geometric (AG) codes of genus 0 curves (i.e. very special MDS codes). A subfield subcode inherits the minimum distance, the automorphism group and encoding and decoding algorithms from its parent supercode. We have previously used the Gröbner bases approach in [11] to find results regarding the parameters and decoding of subfield subcodes of Hermitian curves. Among such codes we also found codes which are optimal or best known. Now we generalize these results to subfield subcodes of extended Norm-Trace codes. In this article we provide a Gröbener basis framework to study these codes. With a Gröbner basis for the set of F q r -rational points of an extended Norm-Trace curve we prove results that help us give explicit algorithms for computing the parameters of subfield subcodes of extended Norm-Trace codes. These results are easily adaptable to encoding and decoding. We have implemented these algorithms in symbolic software.
We finish with some subfield subcodes of extended Norm-Trace codes which either have optimal parameters or are as good as any known code. Their optimality is deduced from [8].
We fix q = p l a prime power and r > 1 a positive integer. In addition t is a prime power power such that F t ⊆ F q r . We denote the trace function of F t over F q r by T F q r /Ft .

Subfield subcodes
For the material in this section, we refer the reader to [12]. Definition 1. Let C be a code over F q r of length n. The subfield subcode of C is defined as C|F t := C ∩ F n t . The trace code of C is defined as Both C|F t and T F q r /Ft (C) are linear codes over F t of length n. In fact, Delsarte's Theorem shows how to calculate the dual of a subfield code, using trace codes: . Let C be a linear code over F q r . Then The map x → x t is an automorphism of F q r which fixes F t pointwise. In fact, this automorphism generates the Galois group of F q r over F t . One can extend this map to a linear space as follows: 12]). Let C be a code over F q r of length n. Define . . , c t n )|c ∈ C}. Stichtenoth in [12] showed that C|F t and T F q r /Ft (C) have the same parameters when considered as codes over the superfield F q r . Proposition 2 ( [12]). Suppose that q r = t m . Let C be a code over F q r . Define . Then C o is the F q r -linear code spanned by C|F t over F q r and C ∧ the F q r -linear code spanned by T F q r /Ft (C) over F q r . Moreover C o and C|F t have the dimension and minimum distance. The codes C ∧ and T F q r /Ft (C) also have the same dimension and minimum distance.

Gröbner bases and affine variety codes
In this section we give the theory of Gröbner bases and discuss how it relates to polynomial functions over a set V . In order to simplify our notation, we will focus on the case of two variables, although the theory applies to any finite set of variables. We suppose that X and Y are two variables and that F denotes a field.
Definition 4 ([2, Chapter 2, Section 2, Definition 7, page 59], [7]). Let ≺ be a monomial order in Definition 6 ([2, Chapter 1, Section 4, Definition 5, page 32]). Let V be a subset of points of F 2 . The ideal of V is defined as We can relate the polynomial functions on the points of V to the ideal I V and the Normal Basis ∆ ≺ (I V ).
The following bound is known as the footprint bound.
Proposition 5. Let V = {P 1 , P 2 , . . . , P n } be a set of points of F 2 . Suppose that ≺ is a monomial order in F[X, Y ]. Let L be a linear subspace of F[X, Y ]. The kernel of the map ev V (f ) : L → F n , which maps the polynomial f ∈ L to the point (f (P 1 ), f (P 2 ), . . . , f (P n )) Pi∈V , is a linear map whose kernel is I V ∩ L.
Proof. Clearly, if f ∈ I V ∩ L, then it belongs to the kernel of ev V . Suppose ev V (f ) is equal to the zero vector. A polynomial f ∈ L evaluates to 0 over each point of V if and only if f ∈ I V . Therefore ev V (f ) evaluates to 0 implies that f ∈ I V . Definition 7 ( [3]). Suppose that V = {P 1 , P 2 , . . . , P n } ⊆ F m q r is a finite set and L is an F q r linear subspace of polynomials. Then the affine variety code is defined as Note that Proposition 5 states precisely that dim In order to understand the subfield subcode construction better, we state a simple way of expressing the Trace code of an affine variety code.
Lemma 8. Suppose that V = {P 1 , P 2 , . . . , P n } ⊆ F m q r is a finite set and L is an F q r linear subspace of polynomials. Then Proof. Let c ∈ T F q r /Ft (C(V, L)). Then

Codes on curves related to the Norm-Trace curve
We start with the codes in this section as an example to help the reader. The codes in this section are generalization of Norm-Trace codes as found in [5]. These codes are a special case of the extended Norm-Trace codes presented in [1]. We fix q a prime power and r an integer. In this case, we let u is the Norm-Trace curve.
Definition 10. The (q r−1 , u)-weight of a monomial is defined as We denote by Definition 11. The extended Norm-Trace code of weight s is Proposition 6 ([1]). The following equality,

holds.
Our aim is to find or bound the dimension of N T u (s)|F t .
Proof. Delsarte's theorem implies that (C|F t ) ⊥ = T F q r /Ft (C ⊥ ) for any code C.
Now the Lemma follows easily.
As N T u (s) is an affine variety code, T F q r /Ft (N T u (s)) is also an affine variety code. Our aim is to find a bound for dim(T F q r /Ft (N T u (s))). In order to determine the kernel of the evaluation of the functions in ) on the points of N T u , we compute a Gröbner basis for the ideal corresponding to N T u .
Lemma 13. The ideal of polynomial functions which vanish on N T u is generated by This is a particular case of Theorem 19. We refer the reader to the proof therein. The next proposition is an immediate consequence of the previous result and Proposition 3.
Corollary 15. If the (q r−1 , u)-weights of the monomials of the nonzero terms of f are congruent mod (q − 1)u, then the (q r−1 , u)-weights of the monomials of the nonzero terms of f mod Since each division preserves the (q r−1 , u)-weights mod q r − 1 and the remainder of the sum equals the sum of the remainders, the corollary follows.
In [9] the authors used simpler techniques to prove that Lemma 14 and Corollary 15 hold for Norm-Trace curves. As there are fewer monomials, but a similar structure remains, the computations will be faster for extended Norm-Trace codes as the divisor u decreases. As the authors state in [1], we can find a lower bound of the minimum distance of N T u (s) from the theory of order domain codes. We use the following minimum distance bound from [6].

Extended Norm-Trace codes
The extended Norm-Trace curve and their associated linear codes were introduced in [1]. Our definition varies slightly from the definition therein. First we need some facts about linearized polynomials. In order to simplify our statement of the definitions, we shall assume L(Y ) is a particular linearized polynomial defined as follows: It can be shown that L is a linear transformation on any extension field F q m of F q r . In fact, the set D L is the image of L as a linear transformation from F q r to F q r . From the definition of A L it is clear that the polynomial A L (L(Y )) evaluates to 0 over F q r . We say that A L annihilates L over F q r . The next proposition shows A L is a linearized polynomial as well.
Proposition 9 ([10, Theorem 3.52, page 101]). Let L be a linearized polynomial over F q r and let D be an F q -linear subspace of F q r . The polynomial is a separable linearized polynomial.  When u = q r −1 q−1 and L(Y ) = T F q r /Fq (Y ) the curve is a Norm-Trace curve. When r = 2 and d = 1 the curve is equivalent to a quasi-Hermitian curve. In [1] the authors also give an additional condition on u. Namely, that there exists v | q r − 1 such that u | v and that all v powers of the elements of F q r are contained in D L = {L(β) | β ∈ F q r }. This is done to find the exact number of points in N T u,L (F q r ).
We shall consider a more general case. Our approach will be to bound or estimate the number of F q r rational points by considering the polynomial A L (X u ).
Lemma 18. Let L be a linearized, separable polynomial in F q r of degree q d where {β ∈ F q r | L(β) = 0} is an F q -linear subspace of F q r of dimension d. Suppose that u|q r − 1 and suppose that h(X) is the greatest common divisor of A L (X u ) and X q r − X. Then the number of F q r -rational points of Proof. Let L(Y ) be as in the hypothesis. By the definition of A L , the elements {L(β) | β ∈ F q r } are precisely the roots of A L (Y ). Suppose that x ∈ F q r and y ∈ F q r satisfy x u = L(y). Then x u ∈ {L(β) | β ∈ F q r }. Therefore A L (x u ) = 0. As L(Y ) is an q d -to-1 function, for each root, x, of h(X) there are q d elements y ∈ F q r such that x u = L(y). Thus, there are deg(h)q d solutions to X u − L(Y ).
Note that X u |A L (X u ). Therefore gcd(A L (X u ), In the next theorem, we find a good basis for the ideal I N T u,L (F q r ) . We prove this by stating that a certain ideal is contained in I N T u,L (F q r ) . Then we prove that both ideals have the same footprint, which implies equality.
Theorem 19. Let L be a linearized, separable polynomial in F q r of degree q d where {β ∈ F q r | L(β) = 0} is an F q -linear subspace of F q r of dimension d. Suppose that u|q r − 1 and that h(X) is the greatest common divisor of A L (X u ) and X q r − X. Then I N T u,L (F q r ) is generated by X u − L(Y ) and h(X).
Proof. Let J = X u − L(Y ), h(X) . Lemma 18 implies the polynomials X u − L(Y ) and h(X) vanish on the points in N T u,L . Therefore #N T u,L ≤ ∆ ≺ (J). Let ≺ be a monomial order where X u ≺ Y q d . As X deg h and Y q d are leading terms of polynomials in J, which is what we wanted to prove.
Definition 20. For any nonnegative integer s, the extended Norm-Trace code of weight s is defined as N T u,L (s) := C(N T u,L (F q r ), M u,q d (s)).

Dimension of subfield subcodes of extended Norm-Trace codes
We aim to either find the exact value or bound the dimension of N T u,L (s)|F t . As N T u,L (s) is an affine variety code, Lemma 8 implies that T F q r /Ft (N T u,L (s)) is also an affine variety code. Our aim is to find a bound for dim Ft (T F q r /Ft (N T u,L (s))). We shall determine the kernel of the evaluation of the functions in on the points of N T u,L . Luckily, we have already computed a Gröbner basis for the ideal of polynomial functions which vanish on N T u,L (F q r ).
Theorem 21. Let L be a linearized, separable polynomial in F q r of degree q d where {β ∈ F q r | L(β) = 0} is an F q -linear subspace of F q r of dimension d. Suppose that u|q r − 1 and that h(X) is the greatest common divisor of A L (X u ) and X q r − X. Let f (X, Y ) be a polynomial spanned by the monomials in M u,q d (s). Suppose that g is the remainder of f after dividing by h(X) and X u −L(Y ) under a monomial ordering where Y q d is the leading monomial of X u − L(Y ). Then g(X, Y ) ∈ M u,q d (s).
Proof. If one divides the monomial X i Y j by X u − L(Y ), the leading term is X i+lu Y j−lq d where d is the quotient when dividing j by q d . Note that the monomials X i Y j and X i+lu Y j−lq d have the same (q d , u)-weight. Thus polynomial division by X u − L(Y ) also preserves the (q d , u)-weight. Clearly when one divides any polynomial by the univariate polynomial h(X), the (q d , u)-weight may not increase.
Extending the argument of [1], we can find a lower bound of the minimum distance of N T u,L (s) from the theory of order domain codes. We use the following minimum distance bound from [6]. Proposition 11 ([6]). Let L be a linearized, separable polynomial in F q r of degree q d where {β ∈ F q r | L(β) = 0} is an F q -linear subspace of F q r of dimension d. Suppose that u|q r − 1 and that h(X) is the greatest common divisor of A L (X u ) and X q r − X. Denote by ∆ the footprint of I N T u,L (F q r ) under lexicographical order where Y > X. For a nonnegative integer s, let M u,q d (s) = M u,q d (s) ∩ ∆. Then the minimum distance of N T u,L (s) satisfies: And the minimum distance of N T u,L (s) ⊥ satisfies: This lower bound is a BCH-like bound. We use this lower bound to estimate the minimum distance of some subfield subcodes of extended Norm-Trace codes as in the following examples .

Search for good subfield subcodes of extended Norm-Trace codes
In order to find the parameters of N T u,L (s) ⊥ |F t , we use the minimum distance of the supercode N T u,L (s) ⊥ itself as a lower bound on the minimum distance of the subfield subcode. The reason we have chosen to work with the subfield subcode of the dual code, N T u,L (s) ⊥ |F t rather than with the subfield subcode of the primary code N T u,L (s)|F t is that it is quite difficult to work with N T u,L (s)|F t directly. Delsarte's theorem allows us to find the dimension of T F q r /Ft N T u (s ), which can be done by looking at the different t powers of the monomials in M u,q d (s ) reduced modulo the ideal of N T u,L . 7.1. The case L(Y ) = T F q r /Fq (Y ) and u| q r −1 q−1 over the field F q r . In this case, The greatest common divisor of A L (X u ) and X q r − X is A L (X u )/X u−1 = X u(q−1)+1 − X. Therefore the length of the codes is n = q r−1 (u(q − 1) + 1). 7.1.1. Binary subcodes. First we consider the extended Norm-Trace curve given by u = 3 and L(Y ) = T F16/F2 over F 16 . We would like to find dim N T 3,T F 16 /F 2 (36)|F 2 . In this case N T 3, 8 . Now we reduce these monomials modulo N T 3,T F 16 /F 2 , which is the ideal generated by the polynomials X 4 +X and X 3 +Y 8 +Y 4 +Y 2 +Y . In this case X 4 is equivalent to X, X 8 is equivalent to X 2 and Y 8 is equivalent to X 3 +Y 4 +Y 2 +Y . We find that T F16/F2 (M 3,8 (8)) is generated by 1, Y, Y 2 , Y 4 , X 3 , X, X 2 . Thus dim T F16/F2 (N T 3,T F 16 /F 2 (8)) = 7 and dim N T 3,T F 16 /F 2 (36)|F 2 = 25. Following Geil [6], the code N T 3,T F 16 /F 2 (36) has minimum distance at least 3. However as T F16/F2 (N T 3,T F 16 /F 2 (8)) contains the all ones codeword, all codewords of N T 3,T F 16 /F 2 (36)|F 2 have even weight. From the table of Grassl [8] we determine that N T 3,T F 16 /F 2 (36)|F 2 is an optimal [32, 25, 4] binary code. Now we consider the extended Norm-Trace curve given by L(Y ) = T F16/F2 and u = 5 over F 16 The code N T 5,T F 16 /F 2 (65) has parameters [48, 44, 3] over F 16 . The dual code is N T 5,T F 16 /F 2 (65) ⊥ = N T 5 (10). In this case M 5,8 (10) = {1, Y, Y 2 , X}.
The set M 5,8 (10) ∧ is spanned by the monomials 1, Y, Y 2 , Y 4 , Y 8 , X, X 2 , X 4 , X 8 . By considering reductions modulo X 6 + X and X 5 + Y 8 + Y 4 + Y 2 + Y we find that Y 8 is equivalent to X 5 + Y 4 + Y 2 + Y and X 8 is equivalent to X 3 . Therefore the evaluation of Y 8 is a linear combination of the evaluation of the remaining monomials in M 5,8 (10) ∧ and we obtain that T F16/F2 (M 5,8 (10)) is generated by {1, Y, Y 2 , Y 4 , X, X 2 , X 4 , X 5 }. Thus dim T r(N T 5,T F 16 /F 2 (10)) = 8 and dim N T 5,T F 16 /F 2 (65)|F 2 = 40. As in the previous example, the code N T 5,T F 16 /F 2 (65) has minimum distance at least 3. However as T r(N T 5,T F 16 /F 2 (10)) contains the all ones codeword, the code N T 5,T F 16 /F 2 (65)|F 2 is an optimal [48, 40, 4] binary code. 7.1.2. Quaternary subcodes. For the curve X 5 + Y 8 + Y 4 + Y 2 + Y we will study the codes N T 5,T F 16 /F 2 (60) and N T 5,T F 16 /F 2 (62). Geil's bound on the minimum distance implies that we N T 5,T F 16 /F 2 (60) and N T 5 (4) modulo the ideal N T 5,T F 16 /F 2 one finds that the reductions of the 43 monomials are contained in M 5,8 (60). Therefore the code N T 5,T F 16 /F 2 (60) is invariant under the Frobenius automorphism x → x 4 . This implies that the code N T 5,T F 16 /F 2 (60) is equal to N T 5,T F 16 /F 2 (60) o and therefore there exists a [48, 43, 3] optimal quaternary code. The reductions of the monomials in M 5,T F 16 /F 2 (62) (4) are also contained in M 5 (62). The same argument implies that there is also an optimal [48, 44, 3] quaternary code. 7.2. The case L(Y ) = Y 8 + Y over the field F 64 . This case corresponds to a a subcover of the Hermitian curve over F 64 . In this case A L (Y ) = Y 8 − Y . We shall suppose that u = 3. The greatest common divisor of A L (X 3 ) = X 24 − X 3 and X 64 − X is A L (X 3 )/X 2 = X 22 − X. Therefore the length of the codes is n = 176. The binary and quaternary codes found in this section have optimal minimum distance for their given dimension or have the best known minimum distance [8].
7.3. ENT codes for applications. We find binary subfield subcodes of very large lengths for modern applications--for example to Flash Memories [4]. The binary subfield subcode of the extended Norm-Trace curves, over F 256 yields a binary [11008, 8923, 110] code. These types of codes are needed in several applications (for example, in flash memories see [4]). To use a binary code of comparable parameters, (r, δ) = (8923/11008, 110/11008), one would have to use a code of much larger length, for example a [16383, 15612, 110] BCH code. It is possible to shorten such a BCH code, but that would degrade the parameters while interfering with decoding. Also, there are no corresponding BCH codes that could lead to codes of length 11008 and rate < 5375/11008. Therefore, by our subfield subcodes of extended Norm-Trace codes, we are able to obtain a vast array of codes that have substantial practical applications as well.