Best Known (138−25, 138, s)-Nets in Base 3
(138−25, 138, 688)-Net over F3 — Constructive and digital
Digital (113, 138, 688)-net over F3, using
- t-expansion [i] based on digital (112, 138, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (112, 140, 688)-net over F3, using
(138−25, 138, 3297)-Net over F3 — Digital
Digital (113, 138, 3297)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3138, 3297, F3, 2, 25) (dual of [(3297, 2), 6456, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3138, 6594, F3, 25) (dual of [6594, 6456, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3138, 6595, F3, 25) (dual of [6595, 6457, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(38, 33, F3, 4) (dual of [33, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(24) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3138, 6595, F3, 25) (dual of [6595, 6457, 26]-code), using
- OOA 2-folding [i] based on linear OA(3138, 6594, F3, 25) (dual of [6594, 6456, 26]-code), using
(138−25, 138, 740387)-Net in Base 3 — Upper bound on s
There is no (113, 138, 740388)-net in base 3, because
- 1 times m-reduction [i] would yield (113, 137, 740388)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 232068 601404 476415 979625 439233 692284 509615 001464 103254 179566 080769 > 3137 [i]