Advances in Mathematics of Communications
Volume 4, No. 3, 2010, 399–404
doi:10.3934/amc.2010.4.399
ON THE LEAST COVERING RADIUS OF BINARY LINEAR
CODES OF DIMENSION 6
Tsonka Baicheva and Iliya Bouyukliev
Institute of Mathematics and Informatics, BAS
Bulgarian Academy of Sciences
P.O.Box 323, 5000 Veliko Tarnovo, Bulgaria
(Communicated by Simon Litsyn)
Abstract. In this work the least covering radii of all binary linear codes of
dimension 6 are determined. Codes of dimension up to 6 and lengths up to 15
having the least covering radius are classified and constructions of codes with
R = t2 [n, k] of every length and dimension up to 6 are given. Examples of
using this classification for the construction of codes with the least covering
radius and dimensions greater than 6 are presented.
1. Introduction
In this work we investigate the covering problem in the Hamming space for
binary linear error-correcting codes. More precisely, we search for a code such that
the spheres of radius r centered at the codewords cover the whole n-dimensional
Hamming space. On the other hand, for any code we can determine the smallest
integer with this property and this integer is called the covering radius of the code.
The main characteristic of error-correcting codes is its ability to correct errors
and the minimum distance which measures the number of correctable errors, is
of primary interest. The covering radius is also an important parameter of errorcorrecting codes. When the maximum-likelihood decoding is performed it is the
measure of the largest number of errors in any correctable error pattern. If it is
less than the minimum distance of the code, no vector can be added to the code
without decreasing its minimum distance. Codes with the smallest covering radius
have various applications in data compression, decoding of errors and erasures,
broadcasting in interconnection networks, write-once memories, speech coding, etc.
In the present paper we determine the values of the function t2 [n, 6], i.e. the
smallest covering radius of any binary linear [n, 6] code. An important part of the
determination of the values of t2 [n, 6] is the suggested heuristic algorithm for the
computation of the covering radius of a linear code. We also classify binary codes of
dimensions up to 6 and small lengths which can be used to construct codes having
a specified length and dimension and with a reasonably small covering radius.
2. Some preliminary results
All codes considered in this work will be binary and linear with no zero coordinates. Let Fqn be the n-dimensional vector space over the finite field with q elements.
2000 Mathematics Subject Classification: Primary: 94B05.
Key words and phrases: Binary linear codes, covering radius, least covering radius, classification of codes.
399
c
2010
AIMS-SDU
400
Tsonka Baicheva and Iliya Bouyukliev
A linear code C is a k-dimensional subspace of Fqn . The ball of radius t around a
word y ∈ Fqn is defined by
{x|x ∈ Fqn , d(x, y) ≤ t}.
Then the covering radius R(C) of a code C is defined as the least possible integer
number such that the balls of radius R(C) around the codewords cover the whole
Fqn , i.e.
R(C) = maxn min d(x, c).
x∈Fq c∈C
A coset of the code C defined by the vector x ∈ Fqn is the set
x + C = {x + c | c ∈ C}.
A coset leader of x + C is a vector in x + C of smallest weight. When the code is
linear its covering radius is equal to the weight of the heaviest coset leader.
It is well known that adding a zero coordinate to a code will increase its covering
radius by 1. That is why in this work we will consider only codes without zero
coordinates.
An [n, k] code is called projective if its dual distance is at least 3. The binary
projective codes do not have zero and repeated coordinates.
An interesting problem is the determination of values of the function tq [n, k], the
least value of R(C) when C runs over the class of all linear [n, k] codes over Fq for
a given q. The values of the function t2 [n, k] for codes of dimensions up to 5 are
determined in [5] and [4]. Upper bounds for t2 [n, k] for codes of dimensions 6 and
7 are derived in [5]. More precisely, for dimension 6 it is proved that
n−8
(1)
t2 [n, 6] ≤
, f or n ≥ 18.
2
The concepts of a normal linear code and the amalgamated direct sum (ADS)
construction for normal codes are introduced in [5].
Definition 2.1. [3] Let C be a binary code of length n and covering radius R. For
(i)
(i)
i = 1, . . . , n let C0 (respectively C1 ) denote the set of codewords in which the
i-th coordinate is 0 (respectively 1). The integer
(i)
(i)
N (i) (x) = maxn {d(x, C0 ) + d(x, C1 )}
x∈F2
is called the norm of C with respect to the i-th coordinate and Nmin = mini N (i)
is called the minimum norm of C. (We use the convention that d(x, 0) = ∞.) The
code C has norm N if Nmin ≤ N . A coordinate i for which N (i) < N is called
acceptable with respect to N .
The code C is normal if it has norm 2R + 1.
The results of the investigations on normal codes are summarized in [3] and for
binary codes they are given in the following theorem.
Theorem 2.2. [3] If C is an [n, k] code with n ≤ 15, k ≤ 5 or n − k ≤ 9, then C
is normal.
One of the main reasons for studying normal codes is the amalgamated direct
sum construction where the code has one fewer coordinate than the direct sum, but
the same covering radius and the same redundancy.
Advances in Mathematics of Communications
Volume 4, No. 3 (2010), 399–404
On the least covering radius of binary linear codes of dimension 6
401
Theorem 2.3. [3] Assume that A is a normal binary [nA , kA ]RA code with the
last coordinate acceptable, and B is a normal binary [nB , kB ]RB code with the first
coordinate acceptable. Then their amalgamated direct sum
˙ = {(a, 0, b)|(a, 0) ∈ A, (0, b) ∈ B}∪
A⊕B
∪{(a, 1, b)|(a, 1) ∈ A, (1, b) ∈ B}
is an [nA + nB − 1, kA + kB − 1]R code with R ≤ RA + RB . More generally, if
the norm of A with respect to the last coordinate is NA and the norm of B with
˙ has norm NA + NB − 1
respect to the first coordinate is NB , then the code A⊕B
1
and hence covering radius at most (NA + NB − 1). In particular, if the covering
2
˙ equals RA +RB , then A⊕B
˙ is normal and the overlapping coordinate
radius of A⊕B
is acceptable.
The following result from [5] is very useful in constructing codes with small
covering radii.
Theorem 2.4. If C is an [n, k]R normal code, there are [n + 2i, k]R + i normal
codes for all i ≥ 0.
3. A heuristic algorithm for computing a lower bound on the
covering radius of a linear code
In the proof of Theorem 4.1 from the next section we use a computer to show
the nonexistence of codes of lengths 22 ≤ n ≤ 54, n odd, and given covering radius.
There are 236779414 such codes and if we try to determine their covering radii
using one of the known for us algorithms it would take years. Here we present a
heuristic algorithm which alow us to show the nonexistence of an [n, k]R code C in
a reasonable time.
The idea of the algorithm is to find as fast as possible a coset leader of weight
greater than R, which means that the covering radius of the code is also greater
than R. It starts with randomly choosing a vector c from a coset Kc = {c + C}. If
the weight of Kc is greater than R, we are done. Otherwise we form N (c), the set of
neighbors of c, which differ from c in one coordinate and use the evaluation function
f (c) to find the current best solution. The function f (c) = wt(Kc )2k − A(Kc )
depends on the weight of the coset wt(Kc ) and the number A(Kc ) of vectors of
minimum weight in Kc . Our aim is to maximize f (c), i.e. to minimize the number
of vectors of minimum weight in the coset. We search in the set of neighbors N (c)
until we obtain a coset of weight greater than R. If the search is not successful, we
add some noise to c by inverting an arbitrary chosen coordinate of its, and repeat
the procedure. The sketch of the algorithm is given below.
Algorithm. LowerBoundCoveringRadius(Rmin )
c, c′ : vector; br0 ,br: integer;
{ br0 := 0;
while br0 < const0
{ br0 := br0 + 1; br := 0;
Select a random vector c ∈ F2n \ {C};
while br < const
{ br := br + 1;
while exists c′ ∈ N (c) such that f (c′ ) > f (c) do
if wt(Kc ) > Rmin break; output R > Rmin
Advances in Mathematics of Communications
c := c′ ;
Volume 4, No. 3 (2010), 399–404
402
Tsonka Baicheva and Iliya Bouyukliev
Add some noise to c;
}
}
}
4. Least covering radius of binary linear codes of dimension 6
The results about the least covering radius of binary linear codes are summarized
in Table 7.1 from [3] where the exact values or bounds for t2 [n, k] for codes of
lengths up to 64 are given. Some improvements of these results are obtained in [1].
In particular, for dimension 6 it is shown that t2 [17, 6] = 5.
Now we will show that the bound (1) for codes of dimension 6 and lengths greater
than 17 is sharp. This way we determine the values of the function t2 [n, 6] for any
length. The approach we use is similar to the approach from [5] and is based on
the determination of the covering radii of the projective codes of dimension 6. We
will note that the covering radii of the binary projective codes of dimensions up to
5 are determined in [7] and [6].
Theorem 4.1.
t2 [n, 6] =
n−8
, f or n ≥ 18.
2
Proof. For codes of lengths 18–21, values of t2 [n, 6] are known and they fulfill the
condition of the theorem. For the rest of the codes of lengths up to 64 the upper
bounds from Table I coincide with the value given in the theorem. What remains
is to prove that these upper bounds are sharp. Let us consider the first open case,
[22, 6] codes. If a [22, 6] code C contains a repeated coordinate, then R(C) ≥
t2 [20, 6] + 1 = 7. Thus, if there exits a [22, 6] code of covering radius 6, it must be
a projective one. Classification of all binary projective codes of dimension up to 6
is done in [2]. We use the results of this classification where 2852541 nonequivalent
binary [22, 6] codes are found, and the heuristic algorithm from the previous section
to show that covering radii af all the codes are greater than 6. Therefore t2 [22, 6] =
7. Let now C be a [24, 6] code. If there are [24, 6] codes of covering radius 7,
they must be projective and by the heuristic algorithm we show that all 8239576
such codes have covering radii greater than 7. Thus t2 [24, 6] = 8 and as t2 [25, 6] ≥
t2 [24, 6] we get t2 [25, 6] = 8. We repeat this until t2 [54, 6] = t2 [55, 6] = 23.
Let C be a [56, 6] code. If it contains a repeated coordinate, then R(C) ≥
t2 [54, 6] + 1 = 23 + 1 = 24. Otherwise, C is a shortened in 7 positions version of
the [63, 6] Simplex code with covering radius 31 and thus R(C) ≥ 31 − 7 = 24.
Therefore t2 [56, 6] = 24 and t2 [57, 6] = 24. Similarly t2 [58, 6] = t2 [59, 6] = 25,
t2 [60, 6] = t2 [61, 6] = 26 and t2 [62, 6] = t2 [63, 6] = 27.
For n ≥ 64, every [n, 6] code must contain a repeated coordinate and t2 [n, 6] ≥
t2 [n − 2, 6] + 1, which implies t2 [n, 6] ≥ ⌊(n − 8)/2⌋ for all n. Therefore the upper
bound is sharp, which completes the proof.
5. Classifications and constructions of binary codes with the least
covering radius
From a practical point of view it is very important not only to know the least value
of the covering radius of a code with given parameters, but to have a construction of
Advances in Mathematics of Communications
Volume 4, No. 3 (2010), 399–404
On the least covering radius of binary linear codes of dimension 6
403
Table 1. Binary codes with R = t2 [n, k] for n ≤ 15 and k ≤ 6
Code
[3, 1]1
[4, 1]2
[5, 1]2
[6, 1]3
[7, 1]3
[8, 1]4
[9, 1]4
[10, 1]5
[11, 1]5
[12, 1]6
[13, 1]6
[14, 1]7
[15, 1]7
[5, 4]1
[6, 4]1
[7, 4]1
[8, 4]2
[9, 4]2
[10, 4]3
[11, 4]3
[12, 4]4
[13, 4]4
[14, 4]5
[15, 4]5
Number
1
1
1
1
1
1
1
1
1
1
1
1
1
4
7
1
37
2
164
5
653
11
2334
25
Code
[3, 2]1
[4, 2]1
[5, 2]2
[6, 2]2
[7, 2]3
[8, 2]3
[9, 2]4
[10, 2]4
[11, 2]5
[12, 2]5
[13, 2]6
[14, 2]6
[15, 2]7
Number
2
2
4
4
7
6
11
9
15
12
20
16
26
Code
Number
[4, 3]1
[5, 3]1
[6, 3]2
[7, 3]2
[8, 3]3
[9, 3]3
[10, 3]4
[11, 3]4
[12, 3]5
[13, 3]5
[14, 3]6
[15, 3]6
3
4
11
12
31
32
76
72
162
149
316
286
[6, 5]1
[7, 5]1
[8, 5]1
[9, 5]2
[10, 5]2
[11, 5]3
[12, 5]3
[13, 5]4
[14, 5]4
[15, 5]5
5
11
2
102
19
841
129
6527
889
47604
[7, 6]1
[8, 6]1
[9, 6]1
[10, 6]2
[11, 6]2
[12, 6]3
[13, 6]3
[14, 6]3
[15, 6]4
6
16
4
255
100
4126
2101
1
47552
such a code too. Here we give classifications and constructions of codes of dimension
up to 6 and R = t2 [n, k].
According to Theorem 1 all [n, k] codes of lengths up to 15 and dimensions up
to 6 are normal. We classify all such codes and determine their covering radii. In
Table 2 the number of codes of R = t2 [n, k] are given. The case k = 1 is trivial and
is included in the Table only for completeness. The generator matrices of the codes
can be found at http//www.moi.math.bas.bg/∼tsonka.
Taking the ADS of each of the classified codes of dimension up to 5 and the
[2s + 1, 1]s repetition code of the necessary length, we can obtain (according to
Theorem 3) a code of any length greater than 15, dimension up to 5 and covering
radius equal to t2 [n, k].
In [5] two normal codes are constructed. All coordinates of these two codes
are acceptable. These are [14, 6, 5]3 and [19, 6, 7]5 codes. The uniqueness of the
[14, 6, 5]3 code was proved in [1] and also follows from the above classification. The
[19, 6]5 codes must be projective (t2 [19, 5] ≥ t2 [17, 6] + 1 = 6) and we use the
classification from [2] to determine the covering radii of all 366089 projective [19, 6]
codes. It turns out that there is only one code with covering radius 5 and therefore
the [19, 6, 7]5 code is unique. Then starting from the [14, 6, 5]3 or [19, 6, 7]5 codes
and applying the ADS with the [2s + 1, 1]s repetition code we can obtain every
[n, 6]R = t2 [n, 6] code of length greater than 14, except a code of length 17. To
obtain a [17, 6]5 code we can apply the ADS with one of the classified [15, 6]4 normal
codes and the [3, 1]1 repetition code.
Advances in Mathematics of Communications
Volume 4, No. 3 (2010), 399–404
404
Tsonka Baicheva and Iliya Bouyukliev
The normal codes classified in this work, can also be used to construct codes with
the least covering radius and dimensions greater than 6. For example, it is known
that t2 [17, 8] = t2 [18, 8] = 4. If we consider the ADS of [8, 4]2 and [10, 5]2 normal
codes, according to Theorem 2 we obtain a [17, 8]4 normal code. In the same way
the ADS of [9, 4]2 and [10, 5]2 normal codes gives a [18, 8]4 normal code.
Let us now consider the ADS of [7, 4]1 and [14, 6]3 normal codes. The result is a
[20, 9]4 normal code of the least covering radius. A [25, 9]6 normal code of the least
covering radius can be obtained by the ADS of a [7, 4]1 code and the [19, 6]5 code
which is constructed in [5] and is proved to be unique in this work.
References
[1] T. Baicheva and V. Varek, On the least covering radius of binary linear codes with small
lengths, IEEE Trans. Inform. Theory, 49 (2003), 738–740.
[2] I. Bouyukliev, On the binary projective codes wih dimension 6, Disc. Appl. Math., 154 (2006),
1693–1708.
[3] G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, “Covering Codes,” Elsevier Science B.V.,
North-Holland, 1997.
[4] G. Cohen, M. Karpovsky, H. Mattson, Jr. and J. Schatz, Covering radius - survey and recent
results, IEEE Trans. Inform. Theory, 31 (1985), 328–343.
[5] R. L. Graham and N. J. A. Sloane, On the covering radius of codes, IEEE Trans. Inform.
Theory, 31 (1985), 385–401.
[6] K. E. Kilby and N. J. A. Sloane, On the covering radius problem for codes: I Bounds on
normalized covering radius, II Codes of low dimension; normal and abnormal codes, SIAM
J. Algebraic Disc. Methods, 8 (1987), 604–627.
[7] N. J. A. Sloane, A new approach to the covering radius of codes, J. Combin. Theory Ser. A,
42 (1986), 61–86.
Received December 2009; revised February 2010.
E-mail address: [email protected]
E-mail address: [email protected]
Advances in Mathematics of Communications
Volume 4, No. 3 (2010), 399–404