We present a planar hypohamiltonian graph on 42 vertices and (as a corollary) a planar hypotraceable graph on 162 vertices, improving the bounds of Zamfirescu and Zamfirescu and show some other consequences. We also settle the open problem whether there exists a positive integer N, such that for every integer n ≥ N there exists a planar hypohamiltonian/hypotraceable graph on n vertices. ᭧

We give a complete classification of binary linear complementary dual codes of lengths up to 13 and ternary linear complementary dual codes of lengths up to 10.

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s+5, 3, 16s+3; 21s+2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26,3, 19] code. Recently, much work has been done concerning LCD codes for both theoretical and practical reasons (see e.g. [1], [5], [6], [8], [11], [12] and the references given therein). For example, if there is a quaternary Hermitian LCD [n, k, d] code, then there is a maximal entanglement [[n, k, d; n − k]] entanglementassisted quantum error-correcting code (EAQECC for short) (see e.g. [11] and [12]). From this point of view, quaternary Hermitian LCD codes play an important role in the study of maximal entanglement EAQECC's.It is a fundamental problem to determine the largest minimum weight d 4 (n, k) among all quaternary Hermitian LCD [n, k] codes for a given pair (n, k). It was shown that d 4 (n, 2) = ⌊ 4n 5 ⌋ if n ≡ 1, 2, 3 (mod 5) and d 4 (n, 2) = ⌊ 4n 5 ⌋ − 1 if n ≡ 0, 4 (mod 5) for n ≥ 3 in [10] and [12]. In this paper, we give some conditions on the nonexistence of quaternary Hermitian LCD codes with large minimum weights. We give a classification of (unrestricted) quaternary [4r, 3, 3r] codes for r = 9, 10, 12, 13, 14, 16 and quaternary [43, 3, 32] codes. Using the above classification and the classification in [3], we completely determine the largest minimum weight among all quaternary Hermitian LCD codes of dimension 3. In addition, for a positive integer s, a maximal entanglement [[21s + 5, 3, 16s + 3; 21s + 2]] EAQECC is constructed for the first time from a quaternary Hermitian LCD [26,3, 19] code. This paper is organized as follows. In Section 2, we prepare some definitions, notations and basic results used in this paper. In Section 3, we give characterizations of quaternary Hermitian LCD codes. It is shown that there is no quaternary Hermitian LCD [ 4 k −1 3 s, k, 4 k−1 s] code for k ≥ 3 and s ≥ 1 (Theorem 3.3). In addition, if 4(4 k−1 n − 4 k −1 3 α) < k, where k ≥ 3 and 4α − 3n ≥ 1, then there is no quaternary Hermitian LCD [n, k, α] code C with d(C ⊥ H ) ≥ 2, where d(C) denotes the minimum (Hamming) weight of a quaternary code C and C ⊥ H denotes the Hermitian dual code of C. If 4(4 k−1 n − 4 k −1 3 α) ≥ k ≥ 3, where 4α − 3n ≥ 1 and there is no quaternary Hermitian LCD [4(4 k−1 n − 4 k −1 3 α), k, 3(4 k−1 n − 4 k −1 3 α)] code C 0 with d(C ⊥ H 0 ) ≥ 2, then there is no quaternary Hermitian LCD [n, k, α] code C with d(C ⊥ H ) ≥ 2 (Theorem 3.4). In Section 4, from the classification of quaternary codes of dimension 3 by Bouyukliev, Grassl and Varbanov [3], we determine...

It is a fundamental problem to determine the largest minimum weight d 3 (n, k) among all ternary linear complementary dual [n, k] codes. In this note, we determine d 3 (n, 4) for n ≡ 11, 14, 16, 17, 20, 24, 29, 30, 33, 36, 39 (mod 40). We also determine d 3 (n, k), when (n, k) = (121s+17, 5), (364s+13, 6), (364s+18, 6) and (1093s+14, 7), for a nonnegative integer s. In addition, we determine d 3 (n, k) for k = n − 2, n −

We present a cubic planar hypohamiltonian graph on 70 vertices, improving the best known bound of 94 by Thomassen and derive some consequences concerning longest paths and cycles of planar $3$-connected graphs. We also show that cubic planar hypohamiltonian graphs on $n$ vertices exist for every even number $n\geq 86$ and that cubic planar hypotraceable graphs exist on $n$ vertices for every even number $n \geq 356$, settling an open question of Holton and Sheehan.