Best Known (170, 170+27, s)-Nets in Base 4
(170, 170+27, 20179)-Net over F4 — Constructive and digital
Digital (170, 197, 20179)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (154, 181, 20165)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 20165, F4, 27, 27) (dual of [(20165, 27), 544274, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4181, 262146, F4, 27) (dual of [262146, 261965, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 262153, F4, 27) (dual of [262153, 261972, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4181, 262153, F4, 27) (dual of [262153, 261972, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4181, 262146, F4, 27) (dual of [262146, 261965, 28]-code), using
- net defined by OOA [i] based on linear OOA(4181, 20165, F4, 27, 27) (dual of [(20165, 27), 544274, 28]-NRT-code), using
- digital (3, 16, 14)-net over F4, using
(170, 170+27, 178076)-Net over F4 — Digital
Digital (170, 197, 178076)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4197, 178076, F4, 27) (dual of [178076, 177879, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4197, 262164, F4, 27) (dual of [262164, 261967, 28]-code), using
- (u, u+v)-construction [i] based on
- linear OA(416, 19, F4, 13) (dual of [19, 3, 14]-code), using
- 2 times truncation [i] based on linear OA(418, 21, F4, 15) (dual of [21, 3, 16]-code), using
- Simplex code S(3,4) [i]
- 2 times truncation [i] based on linear OA(418, 21, F4, 15) (dual of [21, 3, 16]-code), using
- linear OA(4181, 262145, F4, 27) (dual of [262145, 261964, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(416, 19, F4, 13) (dual of [19, 3, 14]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4197, 262164, F4, 27) (dual of [262164, 261967, 28]-code), using
(170, 170+27, large)-Net in Base 4 — Upper bound on s
There is no (170, 197, large)-net in base 4, because
- 25 times m-reduction [i] would yield (170, 172, large)-net in base 4, but