Lifted polynomials over F16 and their applications to DNA codes

TLDR: A new family of polynomials is introduced which generates reversible codes over a finitefield with sixteenelements(F16 orGF(16)).

Abstract: In this paper, we introduce a new family of polynomials which generates reversible codes over a finitefieldwithsixteenelements(F16 orGF(16)). Wenamethepolynomialsinthisfamilyasliftedpolynomials. Some advantages of lifted polynomials are that they are easy to construct, there are plenty of examples of them and it is easy to determine the dimension of codes generated by them. Furthermore we introduce 4-lifted polynomials which provide a rich source for DNA codes. Also we construct codes over F4 that have the best possible parameters from lifted polynomials. In addition we obtain some reversible codes over F4.

Figures

Table 1: 4-power table

Table 2: [9, 3, 5] reversible complement DNA code

Table 3: some examples for the codes that have the best possible parameters, obtained by lifted polynomials over F4

Full text

Filomat 27:3 (2013), 459–466
DOI 10.2298/FIL1303459O
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Lifted Polynomials Over F
16
and Their Applications to DNA Codes
Elif Segah Oztas, Irfan Siap
Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
Abstract. In this paper, we introduce a new family of polynomials which generates reversible codes over a
finite field with sixteen elements (F
16
or GF(16)). We name the polynomials in this family as lifted polynomials.
Some advantages of lifted polynomials are that they are easy to construct, there are plenty of examples of
them and it is easy to determine the dimension of codes generated by them. Furthermore we introduce
4-lifted polynomials which provide a rich source for DNA codes. Also we construct codes over F
4
that have
the best possible parameters from lifted polynomials. In addition we obtain some reversible codes over F
4
.
1. Introduction
The interest on DNA computing started by the pioneer paper written by Leonard Adleman [3]. Adleman
solved a hard (NP- complete) computational problem by DNA molecules in a test tube. DNA sequences
consist of four bases (nucleotides) that are (A) adenine, (G) guanine, (T) thymine and (C) cytosine. DNA
has two strands that are arranged in an order with a rule that is named as Watson Crick complement
(WCC). Briefly the WCC of A is T and vice versa and the WCC of G is C and vice versa. We describe this
as A
c
= T, T
c
= A, G
c
= C and C
c
= G.
In [2], constructions of DNA codes over the finite field with four elements GF(4) are presented. Examples
that have larger sizes comparing to the previous examples in the literature are constructed therein. Also,
[9, 10, 12] focused on constructing large sets of DNA codewords. In [19], DNA codes over a four element
ring F
2
[u]/⟨u
2
− 1⟩ are considered. In [21], for the first time instead of single DNA bases, double DNA pairs
are matched to a 16 element ring and the algebraic structure of these DNA codes are studied. Hamming
distance constraint, reverse-complement constraint, reverse constraint, fixed GC-content constraint are the
most common constraints used in DNA codes [9, 12, 13, 15]. In [20], stochastic search algorithms are used
to design codewords for DNA computing by Tuplan et al. [5] and [6] present genetic and evolutionary
algorithms from sets of DNA sequences. In [18], cyclic codes over GF(4) are used to construct DNA codes.
But, they constrict their study to only linear reversible cyclic codes over GF(4). In [7] and [8] deletion
similarity distance is used that is different from Hamming distance and more suitable for DNA codes. In
[14], Mansuripur et al. show that DNA molecules can be used as a storage media. In [1], a new approach
is developed that extends the work in [10] and [12] and reverse-complement constraint is added to further
prevent unwanted hybridizations. Studies about DNA computing indicate that DNA computing will be
2010 Mathematics Subject Classification. Primary 92D10; Secondary 11T71, 94B05.
Keywords. Reversible codes; Lifted polynomials; DNA codes.
Received: 29 November 2012; Accepted: 24 February 2012
Communicated by Eberhard Malkowsky
Email addresses: elifsegahoztas@gmail.com (Elif Segah Oztas), isiap@yildiz.edu.tr (Irfan Siap)

E.S. Oztas, I. Siap / Filomat 27:3 (2013), 459–466 460
of much interest in the near future. Hence, DNA Error Correcting Codes will be of great value since DNA
computing is faster and can store more memory than silicon based computing systems.
Reversible codes are useful for DNA structure. Reversible codes were introduced by Massey over GF(q)
(q is a prime power) in [16]. In [17], Muttoo and Lal have studied reversible codes over GF(q) (q is prime).
But, these codes are obtained from a specially constructed parity check matrix. In [4], Das and Tyagi give
the generalized form of parity check matrix to obtain reversible codes both odd and even length.
In this paper, we use a special family of polynomials to construct reversible codes different from [16]
and [17]. The polynomials are called ”lifted polynomials” and ”4-lifted polynomials” which are introduced
by authors and these polynomials satisfy the reversibility over F
16
. By Means of these new families of
polynomials and the construction presented, it is possible to construct DNA codes where some properties
can be controlled. Further, by applying this approach we not only obtain odd length DNA codes which
is the case in [21], but also we obtain even length DNA codes. Also, we provide some examples of linear
codes with best possible parameters from lifted polynomial construction which can be considered as DNA
codes.
The rest of paper is organized as follows. In Section 2, we give some basic notions. In Section 3, we
introduce reversible codes obtained by lifted polynomial. In Section 4 we introduce DNA codes that are
generated by using 4-lifted polynomial over F
16
. In Section 5, there are some applications about the codes
that have the best possible parameters over F
4
. Section 6 concludes the paper.
2. Background
Let F be a finite field. A linear code of length n over F is an F-vector space of F
n
. A cyclic code C of length
n over F is invariant with respect to the right cyclic shift operator that maps a codeword (c
0
, c
1
, ..., c
n−1
) ∈ C
to another codeword (c
n−1
, c
0
, ..., c
n−2
) in C. For each codeword (c
0
, c
1
, ..., c
n−1
), we associate the polynomial
1(x) = c
0
+ c
1
x + ... + c
n−1
x
n−1
where c
i
∈ F. Let
Φ :F[x]/(x
n
− 1) → C
1(x) = c
0
+ c
1
x + ... + c
n−1
x
n−1
→ (c
0
, c
1
, ..., c
n−1
)
(1)
For each codeword c = (c
0
, c
1
, ..., c
n−1
), we define the reverse of c to be c
r
= (c
n−1
, c
n−2
, ..., c
0
).
Definition 2.1. A linear code C of length n over F is said to be reversible if c
r
∈ C for all c ∈ C.
The Hamming distance between codewords c
′
and c
′′
, denoted by H(c
′
, c
′′
), is simply the number of
coordinates in which these two codewords differ.
For each polynomial z(x) = z
0
+ z
1
x + ... + z
r
x
r
with z
r
, 0, the reciprocal of z(x) is defined to be the
polynomial z
∗
(x) = x
r
z(1/x) = z
r
+ z
r−1
x
r−1
+ ... + z
0
x
r
. Consider that deg z
∗
(x) ≤ deg z(x) and if z
0
, 0, then
z(x) and z
∗
(x) always have the same degrees. z(x) is called self-reciprocal if and only if z(x) = z
∗
(x).
Theorem 2.2. [16] The cyclic code generated by a monic polynomial 1(x) is reversible if and only if 1(x) is self-
reciprocal where 1(x)|(x
n
− 1).
3. Reversible codes over F
16
obtained by lifted polynomials
Here, we define lifted polynomials which generate reversible codes under the restriction that the length
n is odd.
Definition 3.1. Let 1(x) = a
0
+ a
1
x + ... + a
t
x
t
be a self reciprocal polynomial over Z
2
and 1(x)|(x
n
− 1) (mod 2) . A
lifted polynomial of 1(x) is denoted by ℓ
1
(x) ∈ F
16
[x] and is defined as follows. If t is odd, then

E.S. Oztas, I. Siap / Filomat 27:3 (2013), 459–466 461
ℓ
1
(x) =
t−1
2

i=0
θ
i
; θ
i
=







β
i
x
i
+ β
i
x
t−i
, a
i
, 0
0 , a
i
= 0
, (2)
if t is even, then
ℓ
1
(x) =
t
2

i=0
θ
i
; θ
i
=











β
i
x
i
+ β
i
x
t−i
, a
i
, 0 , i , t/2
0 , a
i
= 0
β
t
2
x
t
2
, a
i
, 0 , i = t/2
(3)
where β
i
∈ F
16
− {0}. There are many lifted polynomials of 1(x) depending on β
i
. The set of ℓ
1
(x) is denoted by
L
1
(x). The terms x
i
and x
t−i
are called complement pairs.
Example 3.2. Here we present some examples of lifted polynomials. Let n = 15 and
1(x) = 1 + x + x
3
+ x
4
+ x
5
+ x
7
+ x
8
be a self reciprocal polynomial and where 1(x)|(x
15
− 1). We can write some arbitrary lifted polynomials of 1(x) as
follows,
ℓ
1
(x) = α
10
+ αx + α
2
x
3
+ α
5
x
4
+ α
2
x
5
+ αx
7
+ α
10
x
8
and
ℓ
1
(x) = α
5
+ x + α
6
x
3
+ α
2
x
4
+ α
6
x
5
+ x
7
+ α
5
x
8
where α
i
∈ F
16
, i ∈ {0, 1, ..., 14}.
Lemma 3.3. Lifted polynomials are self reciprocal polynomials over F
16
.
Proof. The proof follows easily by observing that the lifted polynomials are obtained by self reciprocal
polynomials over Z
2
by Definition 3.1.
Lemma 3.4. Suppose that a set S consists of vectors and their reverses. Then, the code generated by S as an F-vector
subspace is reversible.
Proof. Suppose that every element in S has its reverse in S. Let S be a spanning set and
S = ⟨c
1
, c
r
1
, c
2
, c
r
2
, ..., c
k
, c
r
k
, c
s
1
, c
s
2
, ..., c
s
m
⟩
where c
i
is codeword and c
s
t
= c
r
s
t
(self reversible) where 1 ≤ t ≤ m. Let C = ⟨S⟩. For every codeword
c =

β
i
c
i
∈ C, since (αc
i
+ βc
j
)
r
= αc
r
i
+ βc
r
j
, we have c
r
=

β
i
c
r
i
∈ C where i, j ∈ {1, 2, ..., k, s
1
, ..., s
m
}. Hence C
is a reversible code.
Remark 3.5. In this paper, the notation ⟨S⟩ will denote the F-vector space generated by the set S. The notation (S)
will stand for the ideal generated by S.
Theorem 3.6. Let ℓ
1
(x) be a lifted polynomial over F
16
of a self reciprocal polynomial 1(x) and 1(x)|(x
n
− 1)(mod2)
with deg(1(x)) = t where n is odd. Let
S = {ℓ
1
(x), xℓ
1
(x), ..., x
n−t−1
ℓ
1
(x)} (4)
and C
l
= ⟨S⟩. If C
l
is a code which is generated by the spanning set S, then C
l
is a reversible F
16
-linear code and it is
shortly denoted by C
l
= ⟨ℓ
1
(x)⟩.

E.S. Oztas, I. Siap / Filomat 27:3 (2013), 459–466 462
Proof. By Lemma 3.4 the reverse of each codeword can be found as follows:








Φ








n−t

j=1
β
i
j
x
i
j
ℓ
1
(x)
















r
= Φ








n−t

j=1
β
i
j
x
n−t−1−i
j
ℓ
1
(x)








(5)
since 0 ≤ i
j
≤ n − t − 1 we have n − t − 1 ≥ n − t − 1 − i
j
≥ 0, where β
i
j
∈ F
16
, i
j
∈ {0, 1, ..., n − t − 1} and Φ is
given in ( 1). The claim follows.
Note that ⟨ℓ
1
(x)⟩ is not an ideal but an F
16
-linear code spanned by S.
We now present an example of a reversible code, by making use of lifted polynomials.
Example 3.7. Let n = 15 and 1(x) = 1 + x + x
3
+ x
4
+ x
5
+ x
7
+ x
8
be a self reciprocal polynomial and 1(x)|(x
15
− 1)
and d = 3 for C = (1(x)). We can choose a lifted polynomial ℓ
1
(x) = α
10
+ αx + α
2
x
3
+ α
5
x
4
+ α
2
x
5
+ αx
7
+ α
10
x
8
.
We know k = 7 (dimension of code C) then we choose this value k for C
l
= ⟨ℓ
1
(x)⟩. Then, C
l
gives a [15, 7, 7]
16
reversible code obtained over F
16
.
In this section, all the statements are suitable for reversible codes over F
4
. Further we present some
example of a reversible code over F
4
= GF(4) = F
2
[x]/(x
2
+ x + 1) which is obtained by a lifted polynomial.
This linear code has the best possible parameters.
Example 3.8. Let n = 9 and 1(x) = 1 + x + x
2
be a self reciprocal polynomial and 1(x)|(x
9
− 1) and d = 2 for
C = (1(x)) over Z
2
. We can choose an arbitrary ℓ
1
(x) = ω + x + ωx
2
over F
4
= {0, 1, ω, ω
2
= 1 + ω}. We know k = 7
(dimension of code C) then we choose this value k for C
l
= ⟨ℓ
1
(x)⟩. Then, C
l
gives a [9, 7, 2]
4
reversible code obtained
over F
4
. This code attains the best possible parameters ([11]).
Example 3.9. Let 1(x) = 1 + x + x
3
+ x
4
+ x
5
+ x
7
+ x
8
and 1|(x
15
− 1)(mod 2). Let ℓ
1
(x) = 1 + ωx + ωx
3
+ ωx
4
+
ωx
5
+ ωx
7
+ x
8
over F
4
. We can represent the spanning set as a matrix. Hence, the generator matrix is

























1 w 0 w w w 0 w 1 0 0 0 0 0 0
0 1 w 0 w w w 0 w 1 0 0 0 0 0
0 0 1 w 0 w w w 0 w 1 0 0 0 0
0 0 0 1 w 0 w w w 0 w 1 0 0 0
0 0 0 0 1 w 0 w w w 0 w 1 0 0
0 0 0 0 0 1 w 0 w w w 0 w 1 0
0 0 0 0 0 0 1 w 0 w w w 0 w 1

























and C
l
= ⟨ℓ
1
(x)⟩ gives a [15, 7, 7]
4
-reversible code. This code has the best possible parameters [11].
4. DNA codes over F
16
from 4-lifted polynomial
In this section, we mainly mention about DNA codes over F
16
. But, there is a reversibility problem
for DNA codes over F
16
. Reversibility problem: Let (α, α
2
, 1) be a codeword and corresponds to ATGCTT
in DNA where α → AT, α
2
→ GC, 1 → TT and α, α
2
, 1 ∈ F
16
. Reverse of (α, α
2
, 1) is (1, α
2
, α). (1, α
2
, α)
corresponds to TTGCAT. But, TTGCAT is not reverse of ATGCTT. Because, reverse of ATGCTT is TTCGTA.
We have solved this problem with 4-lifted polynomial that is introduced by authors.
DNA occur in sequences, represented by sequences of letters from the alphabet S
D
4
= {A, T, G, C}. We
define a DNA code of length 2n to be a set of codewords (α
0
, ..., α
n−1
) where n is odd.
We consider GF(16) = F
16
= F
2
[
x
]
/

x
4
+ x + 1

for DNA and DNA double bases (pairs).
α
i
∈ {AA, AT, AG, AC, TT, TA, TG, TC, GG, GA, GC, GT, CC, CA, CG, CT} = S
D
16
. (6)
The most difficult and interesting problem is to provide a matching between the field elements and DNA
alphabets. This matching should obey the rules and properties of DNA. Here, we accomplish this task by

E.S. Oztas, I. Siap / Filomat 27:3 (2013), 459–466 463
Table 1: 4-power table
double DNA pair F
16
(multiplicative) additive
AA 0
TT α
0
1
AT α
1
α
GC α
2
α
2
AG α
3
α
3
TA α
4
1 + α
CC α
5
α + α
2
AC α
6
α
2
+ α
3
GT α
7
1 + α + α
3
CG α
8
1 + α
2
CA α
9
α + α
3
GG α
10
1 + α + α
2
CT α
11
α + α
2
+ α
3
GA α
12
1 + α + α
2
+ α
3
TG α
13
1 + α
2
+ α
3
TC α
14
1 + α
3
presenting Table 1 which is called the 4-power table that gives a correspondence between DNA and F
16
. In
this table, there is a property that the related DNA double pair of an element of F
16
is reverse of the related
DNA double pair of fourth power of the element of F
16
. For instance, α
2
→ GC and (α
2
)
4
= α
8
→ CG.
The Watson-Crick complement is given by A
c
= T, T
c
= A, C
c
= G, G
c
= C. Hence we use Watson-Crick
complement such that (AA)
c
= TT, ..., (TC)
c
= AG. If α is a codeword, we show α = (α
0
, ..., α
n−1
). We define
the complement of α to be α
c
= (α
c
0
, ..., α
c
n−1
), the reverse-complement of α to be α
rc
= (α
c
n−1
, ..., α
c
0
).
The following definition describes how to match the codewords over the field with DNA codewords.
Definition 4.1. Let C be a code over F
16
of length n and c ∈ C be a codeword where c = (c
0
, c
1
, ..., c
n−1
), and c
i
∈ F
16
.
We define
Θ(c) : C → S
2n
D
4
where (c
0
, c
1
, ..., c
n−1
) → (b
0
, b
1
, ..., b
2n−1
) (7)
where each of c
i
is mapped to coordinate pairs (b
2i
, b
2i+1
) for i = {0, 1, ..., n − 1} defined in Table 1. Hence, Θ(c) =
(b
0
, b
1
, ..., b
2n−1
) is a DNA codeword of Θ(C) where b
j
∈ S
D
4
, for j ∈ {0, 1, ..., 2n − 1}.
For instance, (c
0
, c
1
, c
2
, c
3
) = (α, α
5
, α
6
, α
11
) → (ATCCACCT) = (b
0
, b
1
, b
2
, b
3
, b
4
, b
5
, b
6
, b
7
).
In order to obtain DNA codes, we define and use a new family of the lifted polynomials over F
16
.
Definition 4.2. Let 1(x) be a self reciprocal polynomial over Z
2
and 1(x)|(x
n
− 1) with deg(1(x)) = t. A 4-lifted
polynomial of 1(x) is denoted by l
(4)
1
(x) ∈ F
16
[x]
if t is odd, then
ℓ
(4)
1
(x) =
t−1
2

i=0
τ
i
; τ
i
=







β
v
i
x
i
+ β
4v
i
x
t−i
, a
i
, 0
0 , a
i
= 0
, (8)
if t is even, then
ℓ
(4)
1
(x) =
t
2

i=0
τ
i
; τ
i
=











β
v
i
x
i
+ β
4v
i
x
t−i
, a
i
, 0, i , t/2
0 , a
i
= 0
β
t
2
x
t
2
, a
i
, 0, β
t
2
∈ {0, 1, α
5
, α
10
}, i = t/2
(9)
where β
i
∈ F
16
− {0}, ℓ
(4)
1
(x) is shortly denoted by ℓ
(4)
polynomial of 1(x).

Frequently asked questions

Q1. What are the common constraints used in DNA codes?

Hamming distance constraint, reverse-complement constraint, reverse constraint, fixed GC-content constraint are the most common constraints used in DNA codes [9, 12, 13, 15]. 

Q2. What contributions have the authors mentioned in the paper "Lifted polynomials over f16 and their applications to dna codes" ?

In this paper, the authors introduce a new family of polynomials which generates reversible codes over a finite field with sixteen elements ( F16 or GF ( 16 ) ). Furthermore the authors introduce 4-lifted polynomials which provide a rich source for DNA codes. 

Q3. What are the common constraints used to construct DNA codes?

In [20], stochastic search algorithms are used to design codewords for DNA computing by Tuplan et al. [5] and [6] present genetic and evolutionary algorithms from sets of DNA sequences. 

Q4. What is the proof of the s.theorem?

The authors can apply lifted polynomials to generate reversible codes and reversible DNA codes over F4 where F4 = {0, 1, ω, ω2} and A→ 0,T → 1,C→ ω,G→ ω2. 

Q5. What are the names of the two new families of polynomials?

The polynomials are called ”lifted polynomials” and ”4-lifted polynomials” which are introduced by authors and these polynomials satisfy the reversibility over F16. 

Q6. What is the reversibility of a code?

A cyclic code C of length n over F is invariant with respect to the right cyclic shift operator that maps a codeword (c0, c1, ..., cn−1) ∈ 

Q7. what is the proof of the s.theorem?

Example 4.6. Let C = (1(x)) be a reversible linear code length of 9 over Z2 where 1(x) = 1 + x + x3 + x4 + x6 + x7 is a self reciprocal polynomial. 

Q8. What is the definition of a reversible code?

Reversible codes over F16 obtained by lifted polynomialsHere, the authors define lifted polynomials which generate reversible codes under the restriction that the length n is odd. 

Q9. What is the proof of lifted polynomials?

Suppose that every element in S has its reverse in S. Let S be a spanning set andS = ⟨c1, cr1, c2, cr2, ..., ck, crk, cs1 , cs2 , ..., csm⟩where ci is codeword and cst = crst (self reversible) where 1 ≤ t ≤ m.