acct

1. Nonexistence of some Griesmer codes
of dimension 4 over finite fields
Kazuki Kumegawa
Osaka Prefecture University
(joint work with T. Maruta)

2. Contents
1. Optimal linear codes problem
2. The geometric method
3. Proof of Theorem 2
4. Proof of Theorem 3
5. Remark

3. 1. Optimal linear codes problem
Fq: the field of q elements
Fn
q = {(a1, · · · , an) | ai ∈ Fq}
An [n, k, d]q code C means a k-dimensional
subspace of Fn
q with minimum (Hamming)
weight d = d(C),
d(C) = min{wt(c) | c ∈ C, c = 0},
where wt(c) = |{i | ci = 0}| for c = (c1, · · · , cn).

4. The problem of optimizing one of the param-
eters n, k, d for given the other two is called
”optimal linear codes problem”.
Problem 1. Find nq(k, d), the smallest value
of n for which an [n, k, d]q code exists.
Problem 2. Find dq(n, k), the largest value
of d for which an [n, k, d]q code exists.

5. The Griesmer bound
nq(k, d) ≥ gq(k, d) :=
k−1
i=0





d
qi





where x is a smallest integer ≥ x.
A linear code attaining the Griesmer bound
is called a Griesmer code.
Griesmer codes are optimal.

6. Theorem 1 (Maruta 1997).
nq(4, d) ≥ gq(4, d) + 1
for 2q3 − rq2 − q + 1 ≤ d ≤ 2q3 − rq2
for q > r, r = 3, 4 and for q > 2(r − 1), r ≥ 5.
We prove
Theorem 2.
nq(4, d) ≥ gq(4, d) + 1
for 2q3 − rq2 − q + 1 ≤ d ≤ 2q3 − rq2
for 3 ≤ r ≤ q − q/p, q = ph with p prime.

7. We can improve the valid values of r
as follows:
valid values of r
q Thm 1 Thm 2
7 3,4 3,4,5,6
8 3,4 3,4
9 3,4,5 3,4,5,6
11 3,4,5,6 3, 4, · · · , 10
13 3,4,5,6,7 3, 4, · · · , 12

8. Theorem 3.
nq(4, d) = gq(4, d) + 1
for 2q3 − 3q2 − 2q + 1 ≤ d ≤ 2q3 − 3q2 − q for
q ≥ 7.
Theorem 4.
nq(4, d) ≥ gq(4, d) + 1
for q3/2 − q2 − 2q + 1 ≤ d ≤ q3/2 − q2 − q
for q = 2h, h ≥ 3.

9. 2. The geometric method
PG(r, q): projective space of dim. r over Fq
j-flat: j-dim. projective subspace of PG(r, q)
When j = 0, 1, 2, r − 2, r − 1, a j-flat is called
point, line, plane, secundum, hyperplane (hp
for short), respectively.
Fj := the set of j-flats of PG(r, q)
θj := (qj+1−1)/(q−1) = qj +qj−1+· · ·+q+1

10. C : an [n, k, d]q code generated by G.
Since we would like to find nq(k, d),
we assume that G contains no all-zero-columns.
The columns of G can be considered as a
multiset of n points in Σ = PG(k − 1, q)
denoted also by C.

11. An i-point is a point of Σ which has mul-
tiplicity i in C. Denote by γ0 the maximum
multiplicity of a point from Σ in C and let Ci
be the set of i-points in Σ, 0 ≤ i ≤ γ0.
For ∀S ⊂ Σ we define the multiplicity of S
with respect to C, denoted by mC(S), as
mC(S) =
γ0
i=1
i·|S∩Ci|.

12. Then we obtain the partition
Σ = C0 ∪ C1 ∪ · · · ∪ Cγ0 such that
n = mC(Σ),
n − d = max{mC(π) | π ∈ Fk−2}.
Conversely such a partition of Σ as above
gives an [n, k, d]q code in the natural manner.

13. We define
γj = max{mC(∆) | ∆ ∈ Fj}, 0 ≤ j ≤ k − 2.
When C is Griesmer,
γj =
j
u=0





d
qk−1−u





for 0 ≤ j ≤ k − 2.

14. A line l is an i-line if i = mc(l).
An i-plane, an i-secumdum, and an i-hp are
defined similarly.
ai := |{π ∈ Fk−2 | mC(π) = i}|. The list of
ai’s is the spectrum of C. Note that
ai = An−i/(q − 1) for 0 ≤ i ≤ n − d.

15. Let F be the multiset defined by
mF(S) =
γ0−1
i=0
(γ0 − i) · |S ∩ Ci| for ∀S ⊂ Σ.
F is an (f, m)-minihyper if
f = θk−1 − n,
m = min{mF(π) | π ∈ Fk−2}
= θk−2 − (n − d).

16. A set S of s points in PG(r, q), r ≥ 2, is called
an s-arc if no r + 1 points are on the same
hyperplane.
When q ≥ r, one can take a normal rational
curve as a (q + 1)-arc.
A set L of s (t−1)-flats in a t-flat Π is called
an s-arc of (t − 1)-flats in Π if L forms an
s-arc in the dual space Π∗ of Π.

17. 3. Proof of Theorem 2
Theorem 2.
nq(4, d) ≥ gq(4, d) + 1
for 2q3 − rq2 − q + 1 ≤ d ≤ 2q3 − rq2
for 3 ≤ r ≤ q − q/p, q = ph with p prime.
We prove that there exists no [gq(4, d), 4, d]q
code for d = 2q3 −rq2 −q +1, 3 ≤ r ≤ q −q/p,
q = ph with p prime.

18. Let C be a putative Griesmer
[n = gq(4, d), 4, d = 2q3 − rq2 − q + 1]q code.
Since C is Griesmer,
n = 2q3 − (r − 2)θ2 − (q − 1),
γ0 = 2, γ1 = 2θ1 − r, γ2 = n − d = 2θ2 − rθ1

19. We define
γj = max{mC(∆) | ∆ ∈ Fj}, 0 ≤ j ≤ k − 2.
When C is Griesmer,
γj =
j
u=0





d
qk−1−u





for 0 ≤ j ≤ k − 2.

20. There exists an (n − d)-hp in Σ.
What is a code corresponding to a
(2θ2 − rθ1)-plane in Σ = PG(3, q)?
We investigate the geometric structure of a
(2θ2 − rθ1)-plane ∆ to find its spectrum.

21. 3.1 The spectrum of a (2θ2 − rθ1)-plane
Lemma 1.
Let Π be an i-hp through a t-secundum δ.
For an (n − d)-hyperplane Π0 with
spectrum (τ0, · · · , τγk−3),
τt > 0 holds if i + q(n − d) − n − qt < q.

22. Lemma 2. Let Π be an i-hp through a
t-secundum δ. Then,
t ≤
i + q · (n − d) − n
q
If t = max{mc(δ ) | δ ⊂ Π, δ ∈ Fk−3}, then an
i-hp corresponds to an [i, k − 1, i − t]q code.

23. Let D be the multiset for a (2θ2 − rθ1)-plane
∆ in C. Since 2θ2 − γ2 = rθ1, 2θ1 − γ1 = r,
the multiset F obtained from two copies of
∆ with D deleted, say F = 2∆−D, forms an
(rθ1, r)-minihyper.
Lemma 3 (Hill-Ward, 2007).
Every (r(q +1), r)-minihyper in PG(2, q), q =
ph, p prime, with r ≤ q − q/p is a sum of r
lines.

24. By Lemma 3, F consists of r lines, say l1, · · · , lr
in PG(2, q), i.e., F = l1 + · · · + lr.
Since γ0 = 2, the set of lines {l1, · · · , lr}
forms an r-arc of lines.
Since m(li) = 2θ1 − θ1 − (r − 1) = q − r + 2
and m(l) = 2θ1 − r = γ1 for any line l in ∆
with l = li, we have the spectrum of ∆.
Lemma 4. The spectrum of an (n−d)-plane
is: (τq−r+2, τγ1) = (r, θ2 − r)

25. : 2-point
: 1-point
: 0-point
(2(2θ -

26. Let δ be an i-plane.
From Lemmas 2 and 4, we have
q − r + 2 ≤ i+q+r−3
q ,
i.e., q2 − (r − 1)q − (r − 3) ≤ i.
For i ≤ θ2, δ ∩ ∆ is a (q − r + 2)-line since δ
has no 2-point.
We have i ≤ θ2 − (r − 1).

27. Let i = q2 − uq − (r − 3) + s,
0 ≤ u ≤ r − 2, 0 ≤ s ≤ q − 1.
From Lemma 2, a t-line in the i-plane satis-
fies t ≤ q − u + 1.
If t = q−u+1, then i+q+r−3−qt = s ≤ q−1,
and γ2-plane contains a t-line by Lemma 1,
a contradiction. Hence t ≤ q − u.

28. Considering the lines in δ through a fixed
1-point of δ ∩ ∆,
i ≤ (q−u−1)q+q−r+2 = q2−uq−(r−2) < i,
a contradiction.
Hence i-planes remain only for
q2−(r−1)q−(r−3) ≤ i ≤ q2−(r−2)q−(r−2)
or γ2 − (q − 1) ≤ i ≤ γ2.

29. Suppose ai = 0 for all i ≤ θ2.
Then considering the planes through a given
(q − r + 2)-line on ∆, we have
n ≥ (γ2 − (q − 1) − (q − r + 2))q + γ2
= 2q3 − (r − 2)θ2 + q > n,
a contradiction.
Hence, by Lemma 4, we have
i≤θ2
ai ≥ r.

30. Lemma 5.
Let Π be an i-hp through a t-secundum δ.
Let cj be the number of j-hps through δ other
than Π. Then,
j cj = q,
j(γk−2−j)cj = i+qγk−2−n−qt. · · · · · · (cj)

31. On the other hand, if there exists an i-plane
with
i = q2 − (r − 1)q − (r − 3) + s, 0 ≤ s ≤ q − 1,
then any other j-plane satisfies
j ≥ γ2 − (q − 1)
since the coefiicient of cq2−(r−2)q−(r−2) is q2
and RHS of (cj) = i + q + r − 3 − qt < q2.
This implies
i≤θ2
ai ≤ 1, a contradiction.

32. For r = 3, Theorem 1 tells that
nq(4, d) ≥ gq(4, d) + 1 for
2q3 − 3q2 − q + 1 ≤ d ≤ 2q3 − 3q2 for q > 3.
Theorem 3.
nq(4, d) = gq(4, d) + 1 for
2q3 − 3q2 − 2q + 1 ≤ d ≤ 2q3 − 3q2 − q for
q ≥ 7.
We note that the values of d for r = 3 in
Thm 1 and the values of d in Thm 3 are
consecutive.

33. 4. Proof of Theorem 3
Theorem 3.
nq(4, d) = gq(4, d) + 1
for 2q3 − 3q2 − 2q + 1 ≤ d ≤ 2q3 − 3q2 − q for
q ≥ 7.
It is known that n5(4, d) = g5(4, d) + 1.
We assume q ≥ 7.

34. We first give the construction of a
[gq(4, d) + 1, 4, d = 2q3 − 3q2 − 2q + 1]q code.
Let M be the multiset
M = 2 · PG(3, q) + P − (δ1 + δ2 + δ3 + l),
where P: a point, l: a line, δ1, δ2, δ3 : planes
satisfying δ1 ∩ δ2 ∩ δ3 = P and
l ∩ ((δ1 ∩ δ2) ∪ (δ1 ∩ δ3) ∪ (δ2 ∩ δ3)) = ∅.

35. before
puncturing
after
puncturing

36. Then, M gives a
[gq(4, d) + 1, 4, d = 2q3 − 3q2 − 2q + 1]q code.
Next, we prove the nonexistence of a
[gq(4, d), 4, d]q code for d = 2q3 −3q2 −2q +1,
q ≥ 7.

37. Let C be a putative Griesmer
[n = gq(4, d), 4, d = 2q3 − 3q2 − 2q + 1]q code.
Then n = 2q3 − q2 − 3q − 1,
γ0 = 2, γ1 = 2q − 1, γ2 = n − d = 2q2 − q − 2.
Lemma 6. (Kanazawa-Maruta, 2011)
The spectrum of an (n − d)-plane is:
(A) (τq−1, τ2q−2, τ2q−1) = (3, q + 1, q2 − 3) or
(B) (τq−2, τq−1, τ2q−2, τ2q−1) = (1, 2, q, q2−2).

38. From Lemmas 2 and 6, one can show that
ai = 0 for i /∈ {q2 − 3q − 1, · · ·, q2 − 2q − 5,
q2 − 2q − 1, · · ·, q2 − q − 4, q2 − 1, q2, q2 + q − 1,
2q2 − 3q − 1, · · ·, 2q2 − q − 2}.
We eliminate the possibility of an i-plane for
each i.
We first rule out a possible (q2−3q−1)-plane.

39. Suppose a (q2 − 3q − 1)-plane exists.
Lemma 7. The spectrum satisfies
(1)
n−d
i=0
ai = θk−1
(2)
n−d
i=1
iai = nθk−2
(3)
n−d
i=2
i
2
ai = n
2 θk−3 + qk−2
γ0
s=2
s
2
λs

40. From the equalities (1)-(3),
when γ0 ≤ 2, we have
n−d−1
i=0
(n − d − i)ai =
n − d
2
θk−1
−n(n − d − 1)θk−2 + n
2 θk−3 + qk−2λ2 · · · (4)
j
(2q2 − q − 2 − j)cj = i + q + 1 − qt · · · (cj)
We consider the maximum contribution of
(cj) with i = n − d to LHS of (4).

41. q-1

42. ex) q = 7
q2 − 3q − 1 = 27.
The spectrum (B) of an (n − d)-plane is
(τ5, τ6, τ12, τ13) = (1, 2, 7, 47).
the RHS of (cj) t-line
62 5
55 6
13 12
6 13

43. The maximum possible contributions of cj’s
to the LHS of (4) are
(c27, c89) = (1, 6) for t = 5.
(c76, c86, c89) = (4, 1, 2) for t = 6.
(c76, c89) = (1, 6) for t = 12.
(c83, c89) = (1, 6) for t = 13.
From (4), q2λ2 ≤ 8000 + 1891τ5
+(4·78+3)τ6+78τ12+15τ13,
i.e., λ2 ≤ 240.

44. Lemma 8. Let λs be the number of s-points
in PG(k − 1, q).
Then, λ2 = n − θk−1 + λ0 if γ0 = 2.
We have λ0 ≥ θk−2 − i if i < θk−2.
By Lemma 8,
λ2 ≥ 615 − 400 + (57 − 27) = 245,
a contradiction.
Hence, there exists no 27-plane.

45. If q is odd, we can show from (4) that
λ2 ≤ q3 − 5
2q2 + 5
2q + 9
4.
On the other hand, Lemma 8 yields
λ2 ≥ q3 − 2q2 − 1
2q + 3
2.
From q3 − 2q2 − 1
2q + 3
2 ≤ q3 − 5
2q2 + 5
2q + 9
4,
we have q ≤ 3 +
√
6, which contradicts q ≥ 7.
Hence aq2−3q−1 = 0.

46. We can also show aq2−3q−1 = 0 when q is
even.
Similarly, we can prove the nonexistence of
an i-plane for i /∈ {2q2 −3q−1, ···, 2q2 −q−2},
giving a contradiction by (4).
Hence nq(4, d) = gq(4, d) + 1
for 2q3 − 3q2 − 2q + 1 ≤ d ≤ 2q3 − 3q2 − q for
q ≥ 7.

47. 5.Remark.
Thm 2 can be generalized to the following:
Theorem 5. nq(k, d) ≥ gq(k, d) + 1
for d1 − q + 1 ≤ d ≤ d1 = (k − 2)qk−1 − rqk−2
for k − 1 ≤ r ≤ q − q/p, q = ph with p prime.
Lemma 11 (Landjev-Vandendriessche, 2012).
Every (rθr−1, rθr−2)-minihyper in PG(r, q),
q = ph, p prime, r ≤ q − q/p is a sum of r hps.

48. References
[1] R. Hill and H. Ward, A geometric approach to classifying Griesmer codes,
Des. Codes Cryptogr., 44, 169–196, 2007.
[2] Y. Kageyama and T. Maruta, On the construction of optimal codes over
Fq, preprint.
[3] R. Kanazawa and T. Maruta, On optimal linear codes over F8, Electron.
J. Combin., 18, #P34, 27pp., 2011.
[4] I. Landjev and P. Vandendriessche, A study of (xvt, xvt−1)-minihypers in
PG(t, q), Journal of Combinatorial Theory, Series A 119, 1123-1131, 2012.
[5] T. Maruta, On the minimum length of q-ary linear codes of dimension
four, Discrete Math., 208/209, 427–435, 1999.
[6] T. Maruta, On the nonexistence of q-ary linear codes of dimension five,
Des. Codes Cryptogr., 22, 165-177, 2001.
[7] M. Takenaka, K. Okamoto and T. Maruta, On optimal non-projective
ternary linear codes, Discrete Math., 308, 842–854, 2008.