Twisted Reed-Solomon Codes

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by  Peter Beelen

Twisted Reed–Solomon Codes
Peter Beelen1, Sven Puchinger2, and Johan Rosenkilde né Nielsen1
1Department of Applied Mathematics & Computer Science, Technical University of Denmark, Lyngby, Denmark
2Institute of Communications Engineering, Ulm University, Ulm, Germany
Email: [email protected], [email protected], [email protected]
Abstract—WepresentanewgeneralconstructionofMDScodes Definition 1 ([7]): Let α1,...,αn ∈Fq ∪{∞} be distinct,
over a finite field Fq. We describe two explicit subclasses which k < n, and v1,...,vn ∈ F∗q. The corresponding generalized
contain new MDS codes of length at least q/2 for all values of
Reed–Solomon (GRS) code is defined by
q ≥ 11. Moreover, we show that most of the new codes are not
equivalent to a Reed–Solomon code. CGRS ={[v f(α ),...,v f(α )]: f ∈F [x], degf <k}.
Index Terms—MDS Codes, Reed–Solomon Codes n,k 1 1 n n q
In this setting, for a polynomial f of degree degf < k, the
I. INTRODUCTION quantity f(∞) is defined as a , the coefficient of xk−1 in
k−1
A maximum distance separable (MDS) code C(n,k,d) is a the polynomial f.
linear code attaining the Singleton bound, i.e., d=n−k+1 In case v =1 for all i, the code is called a Reed–Solomon
7 i
[1]. The most prominent MDS codes are generalized Reed– (RS) code. Any non-zero evaluation polynomial f is of degree
1
0Solomon(GRS)codes[2].However,therearemanyotherknown degf < k and hence has at most k − 1 roots among the
2constructions for MDS codes, e.g., based on the equivalent evaluation points α ,...,α in F . If ∞ “evaluates” to zero,
1 n q
n problem of finding n-arcs in projective geometry [3], circulant this just means that degf < k−1 and hence f has at most
amatrices [4], Hankel matrices [5], or extending GRS codes. k−2 roots among the remaining evaluation points. This proves
J Recently, Sheekey [6] introduced a new class of maximum that a GRS code is MDS.

5rank distance codes—which are MDS in terms of the rank In this article, we will construct MDS codes of length n

metric. These codes, Twisted Gabidulin codes, were shown to and dimension k using spaces of polynomials that may contain
]
TbenotequivalenttoGabidulincodes(therank-metricanalogues elements of degree degf ≥ k. To define our codes, we will
Iof Reed–Solomon codes). use the following map.
s. In this paper, we introduce a new construction of Hamming- Definition 2: Let V ⊂ F [X] be a k-dimensional F -linear
q q
cmetric MDS codes, inspired by [6]. The idea is to evaluate subspace. Let α ,...,α ∈ F ∪{∞} be distinct and write
[ 1 n q
polynomials of the form α = [α ,...,α ]. We call α ,...,α the evaluation points.
1 n 1 n
1 Then we define the evaluation map of α on V by
v f(x)=a0+a1x+...+ak−1xk−1+ηahxk−1+t,
5 ev (·):V →Fn, f (cid:55)→[f(α ),...,f(α )].
forsome0≤h<k andt<n−k. Byaprudentchoiceoft,h α q 1 n
9
2and η, we ensure that any such polynomial has at most k−1 Here f(∞) is defined as a , the coefficient of x(cid:96) in the
(cid:96)
1zeroes among the evaluation points, even though their degree polynomial f ∈V, where (cid:96):=maxdeg{f :f ∈V}.
0is larger than k−1. This is enough to ensure that the resulting The evaluation map above is F -linear. This means in
. q
1code is MDS. particular that if for a given V the evaluation map is injective,
0 We single out two explicit subclasses of this construction then the code ev (V) ⊂ Fn will be an F -linear code of
7 α q q
wheretheMDSpropertycanbeaprioriensured. Thesecontain length n and dimension k. Injectivity is immediate if (cid:96) < n.
1
:codes of length up to roughly q/2 for any q, and we show that If V consists of all polynomials of degree strictly less than k,
vfor q ≥11, they contain non-GRS MDS codes. the resulting code is an RS code. For other choices of V, the
i
X The results we obtain are somewhat reminiscent of results resulting code might still be equivalent to an RS code; we use
rin [4], where non-GRS MDS codes were constructed of length the following notion of code equivalence.
aroughly q/2 for even q. However, our construction is very Definition 3: Let C ,C be F -linear [n,k] codes. We say
1 2 q
different,andforsmallvaluesofq weverifiedusingacomputer that C and C are equivalent if there is a permutation π ∈S
1 2 n
that our construction produces codes inequivalent to the ones and v := [v ,...,v ] ∈ (cid:0)F∗(cid:1)n such C = ϕ (C ) where
1 n q 2 π,v 1
mentionedin [4]. More importantly,ourconstruction also gives ϕ is the Hamming-metric isometry
π,v
a very simple way to produce non-GRS MDS codes of length
at least q/2 if q is odd and q ≥11. Besides adding new codes ϕπ,v :Fnq →Fnq, [c1,...,cn](cid:55)→[v1cπ(1),...,vncπ(n)].
to the family of known MDS codes, the new code class might
ItisclearfromtheabovedefinitionthatacodeisaGRScode
be interesting for code-based cryptography. As future work,we
ifandonlyifitisequivalenttoanRScode.Thefollowingwell-
will analyze whether our codes or their subfield subcodes are
known theorem provides an effective tool to decide whether a
suitable for this purpose.
code is equivalent to an RS code.
II. PRELIMINARIES Theorem 1 ([5], [8]): A linear code with generator matrix
G=[I |A] is a GRS code if and only if
In this section, we recall several definitions and known results
for future use in the paper. We denote by F the finite field (i) All entries of A are non-zero.
q
with q elements. (ii) All 2×2 minors of A˜ are non-zero, and
(iii) All 3×3 minors of A˜ are zero, IV. MDS TWISTED CODES
where A˜∈Fkq×n−k is given by A˜ij =A−ij1. In general, twisted codes are not MDS for all parameters
Note that any MDS code has a generator matrix of the form α,k,t,h,η. However, in this section we will describe several
G=[I |A] and that items (i) and (ii) above are satisfied for classes of twisted codes that are always MDS.
this A. Hence the difference between GRS and non-GRS MDS
codes will only become apparent using item (iii). If k <3 or A. (∗)-twisted Codes
n−k <3, the matrix A has no 3×3 minors, so the following If (t,h)=(1,0), it is possible to give a succinct condition on
corollary holds. when the code C (1,0,η,α) is MDS. More precisely, we have
k
Corollary 2: Suppose that k < 3 or n−k < 3. Any MDS the following:
code of length n and dimension k is equivalent to an RS code. Lemma 4: Let k <n, α ,...,α ∈F distinct and η ∈F .
1 n q q
We finishthis section by quoting results on t-sum generators Then the twisted code C (α,1,0,η) is MDS if and only if
k
in abelian groups from [4], [9]. We will apply these results
(cid:89)
to the abelian groups (F∗q,·) and (Fq,+) to analyse several η(−1)k αi (cid:54)=1 ∀I ⊆{1,...,n} s.t. |I|=k. (1)
instances of our code construction in the coming sections. i∈I
Definition 4 ([9]): Let (A,⊕) be a finite abelian group and Proof:ThecodeisMDSifandonlyiftheonlypolynomial
k ∈ N. A subset S ⊂ A is called a k-sum generator of A oftherequiredformwhichhask rootsamongtheα isthezero
i
iaf=fo(cid:76)r akil=l1as∈i. WAe, tdheenroetearbeydMist(ikn,cAt )s1t,h.e.s.m,saklle∈stSintseugcehrstuhcaht pInoltyhneofimrisatl.pLlaectefa=0 (cid:54)=(cid:80)0ki,=−s01inaciexoi+theηraw0ixsek dbeegsufc<hakp,omlyankoinmgiailt.
that any S ⊂A with |S|>M(k,A) is a k-sum generator of impossible thatf has k roots among the α . Ifthere is a subset
i
A. I ⊆{1,...,n} with |I|=k and f(α )=0 for all i∈I, we
i
Lemma 3 ([9, Theorem 3.1]): Let |A|=2r for some r ≥6. can write f =ηa (cid:81) (x−α ) and considering the constant
0 i∈I i
Then, for any 3≤k ≤r−2, term it follows that
M(k,A)=r, 1=η(−1)k(cid:89)α . (2)
i
except if A ∈ {Zm,Z ×Zm−1} for some m > 1 and k ∈ i∈I
2 4 2
{3,r−2} in which case If Condition (1) is satisfied, no such f can exist. Conversely, if
there is an I such that η(−1)k(cid:81) α = 1, then η (cid:54)= 0 and
M(k,A)=r+1. (cid:81) i∈I i
f =η (x−α )∈V has k roots among the α , so
i∈I i k,t,h,η i
This lemma suffices for our purposes. See [4], [9] for more the code is not MDS.
information in case |A| is odd. Thisleadstoourfirstexplicitsubclassoftwistedcodeswhich
are MDS1:
III. TWISTED REED–SOLOMON CODES Definition 7: C (α,1,0,η) is a (∗)-twisted code if the
k
Inthissection,wepresentournewcodeconstruction.Similarto elements of α are a subset of G ∪ {0}, for G a proper
RScodes,weevaluatepolynomialswhosefirstkcoefficientswe subgroup of (F∗,·), and if (−1)kη−1 ∈ F∗ \ G. We write
q q
can choose arbitrarily. The difference is that we allow another C∗(α,η):=C (α,1,0,η).
k k
monomialofdegreelargerthank−1tooccurinthepolynomials Theorem 5: Any (∗)-twisted code is an MDS code.
as well. We define a set of evaluation polynomials as follows. Proof: If a (∗)-twisted code C∗(α,η) is not MDS, then
k
Definition 5: Let k,t,h ∈ N such that 0 ≤ h < k ≤ q and Lemma 4 implies that there exists I ⊆ {1,...,n} such that
let η ∈F \{0}. Then, we define the set of (k,t,h,η)-twisted (−1)kη(cid:81) α =1. Since the α are contained in a subgroup
q i∈I i i
polynomials by G of F∗, we have (cid:81) α ∈G, implying that (−1)kη ∈G as
q i∈I i
(cid:40) k−1 (cid:41) well. This gives a contradiction.
V = f =(cid:88)a xi+ηa xk−1+t :a ∈F , Corollary 6: Let Fq be a finite field and let p be the smallest
k,t,h,η i h i q
prime divisor of q−1. Then there exists a (∗)-twisted code of
i=0
length
where we call h the hook and t the twist.
Note that Vk,t,h,η ⊆ Fnq is a k-dimensional Fq-linear n= q−p1 +1.
subspace. Using the evaluation map from Definition 2 for
V = V , we obtain the codes that we will study in this In particular, if q is odd, (∗)-twisted codes can have length
k,t,h,η
article. n= q+1.
2
Definition6:Letα ,...,α ∈F ∪{∞}bedistinctandwrite Proof: The maximum cardinality of a proper subgroup G
1 n q
α = [α1,...,αn]. Let k,t,h,η be chosen as in Definition 5 of F∗q is (q−1)/p. Now let α be G∪{0} in some order in
suchthatk <nandt≤n−k. Then,thecorrespondingtwisted Definition 7.
Reed–Solomon code of length n and dimension k is given by Foroddq (∗)-twistedcodescanthereforeberatherlong. But
the choice of α is very limited in Definition 7, and perhaps
Ck(α,t,h,η)=evα(Vk,t,h,η)⊆Fnq. longer codes could be constructed with (t,h) = (1,0). The
following answers this negatively, by using k-sum generators.
For brevity, we will use the phrase twisted codes rather
Lemma 7: Let η ∈F∗ and let S :={α ,...,α }⊆F∗ be a
than twisted Reed–Solomon codes from now on. Note that q 1 n q
k-sumgeneratorof(F∗,·). ThenthetwistedcodeC (α,1,0,η)
C (α,t,h,η) indeed has dimension k since the evaluation map q k
k is not MDS.
is injective: any polynomial f ∈ V satisfies degf ≤
k,t,h,η
k−1+t<n. In principle η couldbe 0 in the above definition,
1ThisclasscanbeseenastheHamming-metricanalogofTwistedGabidulin
but in that case we simply obtain RS codes. codes[6].However,weusedifferenttechniquesforanalyzingourcodes.
Proof: By the definition of a k-sum generator, there is an points preserves the MDS property. However, if a polynomial
(cid:81)
index set I ⊆ {1,...,n} with |I| = k such that α = f ∈V satisfies f(∞)=0, that means its degree is at
i∈I i k,1,k−1,η
(−1)kη−1. Hence C (α,1,0,η) is not MDS by Lemma 4. most k−2.
k
Theorem 8: Let F be a finite field, with q odd. Further let Corollary12:LetF beafinitefieldofcharacteristicp.Then
q q
η ∈ F∗. Then, for any 3 ≤ k ≤ q−1 −2, the length n of a there exists a (+)-twisted code of length
q 2
twisted code C (α,1,0,η) which is MDS, satisfies n≤ q+1.
Proof: Wekknow that (F∗,·) is a cyclic abelian group2of n= pq +1.
q
order |F∗|=q−1, which is even since q is a power of an odd
q In particular, if q is even, (+)-twisted codes can have length
prime. Hence Lemma 3 implies n= q +1.
2
M(k,F∗)= q−1. Proof: The maximum cardinality of a proper subgroup V
q 2 of F is q/p. Now set S ={∞}∪V in Definition 8.
q
Now suppose n > q+21 and α = [α1,...,αn]. Then, S := Lemma 7 and Theorem 8 have a direct analogue as well.
{α1,...,αn}∩F∗q has cardinality For completeness, we state the results, but since the proofs are
extremely similar, we leave these to the reader.
|S|≥n−1> q+1 −1= q−1 =M(k,F∗)
2 2 q Lemma 13: Let η (cid:54)= 0 and S := {α ,...,α } ∈ F be a
1 n q
and is therefore a k-sum generator of F∗. By Lemma 7, the k-sumgeneratorof(F ,+).ThenthetwistedcodeC (α,1,k−
q q k
code C (α,1,0,η) is not MDS for any η (cid:54)=0. 1,η) is not MDS.
k
Remark 9: For even q, the (∗)-twisted codes cannot attain Theorem 14: Let F be a finite field,with q even. Furtherlet
q
length (q+1)/2. However, we determined by computer search η ∈F∗. Then,forany3≤k ≤ q−2,the lengthn ofa twisted
q 2
that for e.g. q =16, there are many [9,k] twisted codes with code C (α,1,k−1,η) which is MDS, satisfies n≤ q +1 if
k 2
(t,h) = (1,0) for other choices of α and η, for k = 3,4,5. 3<k < q −3 and n≤ q +2 if k ∈{3,q −2}.
2 2 2
See also Section VI.
C. General Theory of Twisted Codes
B. (+)-twisted Codes
For general t and h, it is still possible to derive a criterion
While the result in the previous subsection were based on foracodeC (α,t,h,η) tobe MDS. Wedo soin thefollowing
properties of the multiplicative group (F∗q,·), it is also possible lemma, genekralizing Lemmas 4 and 10.
tousethestructureoftheadditivegroup(Fq,+).Thisstructure Lemma 15: Let α1,...,αn ∈ Fq and define for I ⊆
arises when considering the case (t,h)=(1,k−1). We have {1,...,n} the polynomial (cid:80) σ xi := (cid:81) (x − α ). The
i i i∈I i
the following analogue of Lemma 4. twisted code C (α,t,h,η) is MDS if and only if the matrix
Lemma 10: Let k < n ≤ q, α ,...,α ∈ F distinct and k
η ∈ Fq. Then the twisted code C1k(α,1,kn−1,qη) is MDS if η−1−σh−t+1 −σh−t+2 ... −σh 
and only if  σk−1 1 
 
(cid:88)  σk−2 σk−1 1 
η αi (cid:54)=−1 ∀I ⊆{1,...,n} s.t. |I|=k. (3)  σk−3 σk−2 σk−1 1 ,
i∈I  ... ... ... ... ... 
Proof:ThecodeisMDSifandonlyiftheonlypolynomial
σ ... σ σ σ 1
k−t+1 k−3 k−2 k−1
f =a +...+a xk−1+ηa xk ∈V (cid:124) (cid:123)(cid:122) (cid:125)
0 k−1 k−1 k,1,k−1,η
=:AI∈Ftq×t
havingk rootsamongtheα isthezeropolynomial.Iff (cid:54)=0is
i is regular for all I ⊆{1,...,n} such that |I|=k.
suchapolynomial,thereisasubsetI ⊆{1,...,n}with|I|=
k and f(α )=0 for all i∈I. Writing f =ηa (cid:81) (x− Proof: Let f ∈ Vk,t,h,η be a polynomial with at least k
i k−1 i∈I rootsamongtheα ’s.Then,thereisanindexsetI ⊆{1,...,n}
α ), with a (cid:54)=0, we obtain a contradiction by considering i
i k−1 with |I|=k and f(α )=0 forall i∈I. We can factorf into
the coefficient of xk−1 on both sides, since i
f(x)=g(x)·σ(x), with
(cid:88) (cid:88)
α =−ηα α ⇐⇒ η α =−1
k−1 k−1 i i t−1 k
(cid:88) (cid:88) (cid:89)
i∈I i∈I g(x)= g xi, σ(x)= σ xi := (x−α ).
i i i
(cid:80)
Conversely, if there is an I such that η i∈I(−αi)=1, then i=0 i=0 i∈I
(cid:81)
f =η (x−α ) is a polynomial of the appropriate form
i∈I i Note that σ =1. Since by construction, the coefficients in f
k
having k roots among the α , so the code is not MDS.
i to xk,...,xk+t−2 are zero, we obtain the following system of
As in the previous subsection, this naturally gives rise to a
(t−1) equations in the g ’s,
j
subclass of twisted codes.
Definition 8: C (α,1,0,η) is a (+)-twisted code if the i
k (cid:88)
elements of α are a subset of V ∪ {∞}, for V a proper 0= gjσi−j i=k,...,k+t−2, (4)
subgroupof(F ,+),andifη−1 ∈F \V.WewriteC+(η,∗):= j=0
q q k
Ck(α,1,0,η). where gj := 0 for all j ∈/ {0,...,t−1}, and σj := 0 for
Theanalysisofthesecodesfollowsthatoftheirmultiplicative j ∈/ {0,...,k}. Considering the coefficients of xk−1+t and xh,
counterparts from the previous subsection very closely. In we obtain ηa =g and a =(cid:80)h g σ and hence
h t−1 h j=0 j h−j
particular we have the following:
Theorem 11: Any (+)-twisted code is an MDS code. h
(cid:88)
Proof: We simply apply Lemma 10 instead of Lemma 4. 0=η−1gt−1− gjσh−j. (5)
Some care must be taken that adding ∞ to a set of evaluation j=0
Equations (4) and (5) result in a homogeneous system of t Corollary 19: Suppose (F∗,·) has a subgroup G such that
q
equations in t variables g :=[g ,g ,...,g ]T: |F∗\G|>6.Thenforanyn,kwith2<k <n−2andn≤|G|
t−1 t−2 0 q
there exists a non-GRS MDS (∗)-twisted code. Similarly if
A ·g =0 (6)
I (F ,+) has a subgroup V such that |F \V|>6, then for any
q q
The code C (α,t,h,η) is MDS if and only if the only polyno- n,k with 2<k <n−2 and n≤|V| there exists a non-GRS
k
mial f ∈Vk,t,h,η with at least k roots is the zero polynomial, MDS (+)-twisted code.
which holds if and only if the system (6) has only the zero Corollary 20: Let F (cid:40)F with |F \F |>6. Let 2<k <
s q q s
vector as solution for all choices of index sets I ⊆{1,...,n} n−2 and n ≤ s. Then, there exists η ∈ F \F such that
q s
with |I|=k. This implies the claim. C (α,t,h,η) is MDS but not equivalent to a GRS code.
k
Remark 16: If one includes ∞ as evaluation point and h= Remark 21: From q ≥ 13, non-GRS (∗)-twisted or (+)-
k−1, the above lemma is still true when considering I not twisted codes of length (cid:100)(q +1)/2(cid:101) is guaranteed by Corol-
containing ∞. Indeed if f(∞)=0,then degf <k−1,which lary 19. If q = pm is a prime power with p > 3 and m > 1,
means the MDS property is not affected. then the restriction |F \F | > 6 of Corollary 20 is fulfilled
q s
As we will see in Section VI many long MDS codes can be for all F (cid:40)F .
s q
obtained using twisted codes for particular values of q. Hence
an upper bound like in Theorems 8 and 14 does not hold for VI. COMPUTER SEARCHES
general h and t. On the other hand, it seems harder to find In this section, we present exhaustive computer searches
explicit constructions of such long MDS codes. We do have for twisted codes over small field sizes. Since we are most
the following result. interestedinnon-GRScodes,weonlyperformsearchesforη (cid:54)=
Theorem 17: Let Fs (cid:40) Fq be a proper subfield and 0andmin{k,n−k}>2(cf.Corollary2). Thecompututations
α1,...,αn ∈ Fs. If η ∈ Fq \ Fs, then the twisted code were carried out using SageMath v7.4 [10]. The full results
C (α,t,h,η) is MDS. and the source code can be downloaded from http://jsrn.dk/
k
Proof: Let η ∈Fq \Fs. Let I ⊆{1,...,n} be an index code-for-articles.
set with |I| = k and A be the corresponding matrix as in
I
Lemma 15. Using elementary row operations, we can bring A. Number of (∗)-Twisted Codes
AI into lower triangular form with diagonal elements η−1+ We counted all (∗)-twisted codes over Fq for q ≤ 19 and
T,1,...,1 for a certain T ∈ Fq. Using that σi ∈ Fs for all all (+)-twisted codes over Fq for q =16 and q =49, i.e., the
i (since the same holds for all αi), we conclude that in fact numberofsetsSandη’sthatfulfilltheconditionsofDefinition7
T ∈Fs. Since η (cid:54)∈Fs, the diagonal elements of the triangular respectively Definition 8. Moreover, we have determined how
form of AI are all non-zero, implying that AI is regular. manyoftheresultingcodesareinequivalentandwhichofthose
are not GRS codes.
V. NON-GRS MDS TWISTED CODES
As predicted,foroddq,there are (∗)-twistedcodes oflength
Since GRS codes are always MDS and well studied, we will
up to q+1 and arbitrary k. It also turns out that almost all
in this section show that most of the twisted codes are not 2
(∗)-twisted codes are non-GRS. In particular, there is exactly
equivalent to an RS code. The main result is the following
1 (∗)-twisted [11,6,3] code which is not GRS, even though
theorem.
Corollary 2 did not guarantee this.
Theorem 18: Let α ,...,α ∈F and 2<k <n−2. Fur-
1 n q Table I exemplifies the results for q =19.
thermore,letH⊆F satisfythatthetwistedcodeC (α,t,h,η)
q k
is MDS for all η ∈ H. Then there are at most 6 choices of
TableI
η ∈H such that C (α,t,h,η) is equivalent to an RS code.
k NUMBER OF(∗)-TWISTED CODES OVERF19 (TOTAL /
Proof: Since Ck(α,t,h,η) is an MDS code, it has a INEQUIVALENT / NON-GRS).
generator matrix of the form G=[I |A]. Equivalently, there
exist polynomials f(i) ∈ V such that for all i and j in n\k 3 4 5 6 7
k,t,h,η
1,...,k it holds that f(i)(α ) = 1 if i = j and f(i)(α ) = 0 6 1974/73/67
j j
if i (cid:54)= j. Further for j > k we have f(i)(α ) = A . In 7 1092/67/67 1092/63/63
j i,j−k
particular, since f(i) ∈ V , the (i,j)th entry of A is of 8 405/25/25 405/25/25 405/25/25
k,t,h,η
the form A =c +d η for certain c ,d ∈F . 9 90/7/7 90/6/6 90/6/6 90/7/7
i,j i,j i,j i,j i,j q
Nowweuseitem(iii)toderiveanupperboundonthenumber 10 9/2/2 9/1/1 9/1/1 9/2/2 9/1/1
of choices of η such that C (α,t,h,η) is equivalent to an RS
k
code. Let us consider the minor M of the first three rows and
B. Comparison with Roth–Lempel Codes
columns of A˜. Then C (α,t,h,η) is not equivalent to an RS
code if M does not vaknish. However, since A˜ = 1 , Roth and Lempel [4] gave a construction of non-GRS MDS
this minor is of the form i,j ci,j+di,jη codes: given S ⊂ Fq ∪{∞} with n = |S| which is not a
(k−1)-sum generator of (F ,+), it produces an [n,k] MDS
p(η) q
M = , code2. Roth and Lempel point out, similar to our Definition 8,
(cid:81)3i,j=1(ci,j +di,jη) thate.g. subgroups ofFq willgive suchnon-(k−1)-generators.
where p(η) is a polynomial in η of degree at most 6. Hence When q is even, these explicit constructions allow codes in
M can vanish for at most six values of η, which implies the a similar range as (+)-twisted codes. When q is odd, their
theorem. construction is much worse; for q an odd prime, they remark
Theorem 18 directly implies the existence of non-GRS MDS
2We say that a set S containing ∞ is a k-sum generator if S\{∞} is
twistedcodesformanyfieldsizes,asappearsfromthefollowing
a k-sum generator. Note that for comparison with our codes, we relax the
corollaries. constructionof[4]byallowingS whichdoesnotcontain∞.
in [9] that asymptotically their construction has at most length apply the RS decoderq times. This works with errors,erasures,
q(1+o(1)). soft-decision,list-decoding,etc.Inparticular,wehave(cf.[12]):
k
For small q one can exhaustively search for all non-(k−1)-
TableII
sumgenerators,however,andtherebyproduceallRoth–Lempel COUNTING MDS TWISTED CODES
(RL) codes. We have done this for some parameters with the (TOTAL/INEQUIVALENT/NON-GRS. BLANK = 0/0/0)
aim of determining how often (∗)-twisted or(+)-twisted codes
q n k=3 4 5 6 7 8 9
areequivalenttoRLcodes;especiallyforthe(+)-twistedcodes 7 6 38/3/2
where the possible range of parameters largely coincide. 8 6 406/5/4
7 63/2/1
Our computer searches indicate that the code families are 8 14/2/1
9 6 2374/7/5
largely independent. We give three examples:
7 216/3/3 332/3/2
For(q,n,k)=(13,7,3),thereare35inequivalentRLcodes, 8 4/1/1 40/1/1 36/1/1
9 4/1/1 4/1/1
whilethereare2(∗)-twistedcodes;1codeisinbothsets.There 11 6 32518/15/11
are no (+)-twisted codes of these parameters, but there are 8 7 6286/21/19 8554/20/18
8 585/15/15 160/7/6 960/9/7
twisted codes with (t,h)=(1,k−1); 2 of these are RL codes. 9 40/3/3 20/1/0 135/1/0
10 2/1/1 22/2/1
For (q,n,k) = (16,8,5), there are 186 inequivalent RL 13 6 216722/26/21
codes, while there are 9 inequivalent (+)-twisted codes. These 7 71618/80/75 98430/80/75
8 11164/165/160 5176/98/93 26916/139/134
codes are all different. There are 83 twisted codes in total with 9 1110/32/31 41/4/3 381/8/5 5424/24/21
10 138/4/4 93/3/0 1167/4/1
(t,h)=(1,k−1), and 10 of these are RL codes. 11 24/1/1 254/1/0
For(q,n,k)=(23,12,5),therearenoRLcodes,whilethere 12 2/1/1 26/2/1
is 1 equivalence class of (∗)-twisted codes.
Theorem 22: An [n,k] twisted code over F can be decoded
C. Length ≈q/2 Codes with “Exotic Twists” q
up to half the minimum distance in complexity O∼(qn).
(∗)-twisted and (+)-twisted codes are explicit subclasses of
Note that even if the twisted code is not MDS,this approach
the cases (t,h)=(1,0) respectively (t,h)=(1,k−1) which
still gives a list-decoder up to (n − k + 1)/2: collect the
allow codes of length ≈ q/2 for odd respectively even q. A
codewords obtained from each guess of a , and the correct
h
natural question is if similar long MDS codes are possible for
codeword is on the resulting list if the number of errors is at
twisted codes of other (t,h).
most the decoding radius of the RS decoder.
We have no explicit construction, but exhaustive search
indicatesaresounding’yes’: infact,foranyq ≤19weverified VIII. CONCLUSION
that for almost any choice of (t,h) there is an [n,k] twisted
We have introduced twisted Reed–Solomon codes, a new
MDS code for n = (cid:98)q/2(cid:99) and 3 ≤ k ≤ n−3, with the only class of F -linear codes, and demonstrated that for q ≥7 the
q
exceptionsbeing(t,h)=(1,k−1)or(t,h)=(k−1,1)which
class contains manynewnon-GRS MDS codes. We singledout
fails for (q,k)=(17,4) and q =19 and any k. twoexplicitsubclasses,(∗)-twistedand(+)-twistedcodes,with
In the cases (t,h) = (1,k −1) failure should indeed be which we can construct MDS codes of length at least q/2 for
expectedforhighenoughkduetoLemma13andtheasymptotic anyfieldsizeq,andthatformostfieldsizes,mostofthesecodes
upper bounds on M(k,F ) discussed in [9]. We have currently
q will be non-GRS. Using computer searches we demonstrated
no explanation for the “dual” parameters (t,h) = (k−1,1)
that there seems to be many other and longer MDS, non-GRS
failing.
twisted RS codes.
D. Counting Twisted MDS Codes
REFERENCES
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q