An
n
,
k
,
d
q
code is a linear code of length
n
, dimension
k
and minimum weight
d
over the field of order
q
. It is known that the Griesmer bound is attained for all sufficiently large
d
for ...fixed
q
and
k
. We deal with the problem to find
D
q
,
k
, the largest value of
d
such that the Griesmer bound is not attained for fixed
q
and
k
.
D
q
,
k
is already known for the cases
q
≥
k
with
k
=
3
,
4
,
5
and
q
≥
2
k
-
3
with
k
≥
6
, but not known for the case
q
<
k
except for some small
q
and
k
. We show that our conjecture on
D
3
,
k
is valid for
k
≤
9
.
We introduce a new concept “geometric extending” for linear codes over finite fields and consider the extendability of divisible codes. As an application, we construct new Griesmer n,5,dq codes for ...3q4−5q3+1≤d≤3q4−5q3+q2 with q≥3, combining the known geometric methods such as projective dual, geometric extending and geometric puncturing.
We consider the problem of determining n4(5,d), the smallest possible length n for which an n,5,d4 code of minimum distance d over the field of order 4 exists. We prove the nonexistence of ...g4(5,d)+1,5,d4 codes for d=31,47,48,59,60,61,62 and the nonexistence of a g4(5,d),5,d4 code for d=138 using the geometric method through projective geometries, where gq(k,d)=∑i=0k−1d∕qi. This yields to determine the exact values of n4(5,d) for these values of d. We also give the updated table for n4(5,d) for all d except some known cases.
In this paper we generalize the construction of Griesmer codes of Belov type to construct
g
q
(
k
,
d
)
+
t
,
k
,
d
q
codes with an integer
t
≥
1
, where
g
q
(
k
,
d
)
=
∑
i
=
0
k
-
1
d
/
q
i
. ...This leads to the construction of several codes of length
g
q
(
k
,
d
)
+
1
, many of which are optimal. We also construct a
q
-divisible
q
2
+
q
,
5
,
q
2
-
q
q
code through projective geometry. As a projective dual of the code, we construct optimal codes, giving
n
q
(
5
,
d
)
=
g
q
(
5
,
d
)
+
1
for
q
4
-
q
3
-
q
2
+
1
≤
d
≤
q
4
-
q
3
-
2
q
,
q
≥
3
, where
n
q
(
k
,
d
)
is the minimum length
n
for which an
n
,
k
,
d
q
code exists.
We construct Griesmer n,5,dq codes for 2q4−3q3+1≤d≤2q4−3q3+q2 and for 3q4+5q3+1≤d≤3q4+5q3+q2 for every q≥3 using some geometric methods such as projective dual and geometric puncturing.
An algorithm for computing the weight distribution of a linear
n
,
k
code over a finite field
𝔽
q
is developed. The codes are represented by their characteristic vector with respect to a given ...generator matrix and a generator matrix of the
k
-dimensional simplex code
𝑺
q
,
k
.
We construct a lot of new $n,4,d_9$ codes whose lengths are close to the Griesmer bound and prove the nonexistence of some linear codes attaining the Griesmer bound using some geometric techniques ...through projective geometries to determine the exact value of $n_9(4,d)$ or to improve the known bound on $n_9(4,d)$ for given values of $d$, where $n_q(k,d)$ is the minimum length $n$ for which an $n,k,d_q$ code exists. We also give the updated table for $n_9(4,d)$ for all $d$ except some known cases.
We construct Griesmer
n
,
5
,
d
q
codes for
2
q
4
+
1
≤
d
≤
2
q
4
+
q
2
-
q
using some geometric methods such as projective dual and geometric puncturing.
Using an exhaustive computer search, we prove that the number of inequivalent ( 29 , 5 ) -arcs in PG ( 2 , 7 ) is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in ...Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linear codes are related, which enables us to apply an algorithm for classifying the associated linear codes instead. Related to this result, several infinite families of arcs and multiple blocking sets are constructed. Lastly, the relationship between these arcs and the Barlotti arc is explored using a construction that we call transitioning.
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