Best Known (199−27, 199, s)-Nets in Base 4
(199−27, 199, 20182)-Net over F4 — Constructive and digital
Digital (172, 199, 20182)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 18, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-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 (5, 18, 17)-net over F4, using
(199−27, 199, 198965)-Net over F4 — Digital
Digital (172, 199, 198965)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4199, 198965, F4, 27) (dual of [198965, 198766, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, 262217, F4, 27) (dual of [262217, 262018, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- 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(4127, 262145, F4, 19) (dual of [262145, 262018, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(418, 72, F4, 7) (dual of [72, 54, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(418, 85, F4, 7) (dual of [85, 67, 8]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4199, 262217, F4, 27) (dual of [262217, 262018, 28]-code), using
(199−27, 199, large)-Net in Base 4 — Upper bound on s
There is no (172, 199, large)-net in base 4, because
- 25 times m-reduction [i] would yield (172, 174, large)-net in base 4, but