Best Known (125, 141, s)-Nets in Base 4
(125, 141, 524297)-Net over F4 — Constructive and digital
Digital (125, 141, 524297)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (116, 132, 524288)-net over F4, using
- net defined by OOA [i] based on linear OOA(4132, 524288, F4, 16, 16) (dual of [(524288, 16), 8388476, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4132, 4194304, F4, 16) (dual of [4194304, 4194172, 17]-code), using
- 1 times truncation [i] based on linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(4132, 4194304, F4, 16) (dual of [4194304, 4194172, 17]-code), using
- net defined by OOA [i] based on linear OOA(4132, 524288, F4, 16, 16) (dual of [(524288, 16), 8388476, 17]-NRT-code), using
- digital (1, 9, 9)-net over F4, using
(125, 141, 2113169)-Net over F4 — Digital
Digital (125, 141, 2113169)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4141, 2113169, F4, 16) (dual of [2113169, 2113028, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 4194357, F4, 16) (dual of [4194357, 4194216, 17]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(489, 4194305, F4, 11) (dual of [4194305, 4194216, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(48, 52, F4, 4) (dual of [52, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4141, 4194357, F4, 16) (dual of [4194357, 4194216, 17]-code), using
(125, 141, large)-Net in Base 4 — Upper bound on s
There is no (125, 141, large)-net in base 4, because
- 14 times m-reduction [i] would yield (125, 127, large)-net in base 4, but