Best Known (101, 123, s)-Nets in Base 3
(101, 123, 688)-Net over F3 — Constructive and digital
Digital (101, 123, 688)-net over F3, using
- t-expansion [i] based on digital (100, 123, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (100, 124, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (100, 124, 688)-net over F3, using
(101, 123, 3360)-Net over F3 — Digital
Digital (101, 123, 3360)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3123, 3360, F3, 22) (dual of [3360, 3237, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3123, 6597, F3, 22) (dual of [6597, 6474, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3123, 6597, F3, 22) (dual of [6597, 6474, 23]-code), using
(101, 123, 530956)-Net in Base 3 — Upper bound on s
There is no (101, 123, 530957)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 48520 086249 291078 798194 069806 454884 178047 205019 558247 550955 > 3123 [i]