A346003 Distance 3 lexicode over the alphabet {0,1,2}, with the codewords written in base 10.

0, 13, 26, 32, 42, 46, 61, 65, 75, 325, 336, 357, 362, 373, 383, 394, 396, 413, 584, 651, 658, 677, 699, 716, 812, 825, 832, 840, 847, 863, 878, 898, 909, 975, 982, 1001, 1023, 1043, 1048, 1148, 1165, 1170, 1194, 1208, 1223, 1254, 1269, 1330, 1341, 1421, 1452

Offset: 1

Author

N. J. A. Sloane, Jul 20 2021

Keywords

• nonn
• base

Comments

Lexicographically earliest sequence of ternary words such that any two distinct words differ in at least 3 positions.

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..6000
• J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235.
• J. H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Transactions on Information Theory, 32:337-348, 1986.

Crossrefs

Lexicodes of minimal distance 1,2,3,... over alphabets of size 2: A001477, A001969, A075926, A075928, A075931, A075934, ...; size 3: A001477, A346002, A346003; size 10: A001477, A343444, A333568, A346000, A346001.

Programs

Maple

(See A346000).

Python

def t(n):
    d = []
    while n:
        d.append(n%3)
        n //= 3
    return d
def dif(n1, n2):
    return sum(d1 != d2 for d1, d2 in zip(n1 + [0] * (len(n2)-len(n1)), n2))
a = [0]
for n in range(2000):
    if all(dif(t(n1), t(n)) >= 3 for n1 in a):
        a.append(n)
print(a) # Andrey Zabolotskiy, Sep 30 2021

Extensions

Terms a(36) and beyond from Andrey Zabolotskiy, Sep 30 2021