Best Known (137, 137+16, s)-Nets in Base 4
(137, 137+16, 1048584)-Net over F4 — Constructive and digital
Digital (137, 153, 1048584)-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 (128, 144, 1048575)-net over F4, using
- net defined by OOA [i] based on linear OOA(4144, 1048575, F4, 16, 16) (dual of [(1048575, 16), 16777056, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4144, 8388600, F4, 16) (dual of [8388600, 8388456, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(4144, 8388600, F4, 16) (dual of [8388600, 8388456, 17]-code), using
- net defined by OOA [i] based on linear OOA(4144, 1048575, F4, 16, 16) (dual of [(1048575, 16), 16777056, 17]-NRT-code), using
- digital (1, 9, 9)-net over F4, using
(137, 137+16, 6934053)-Net over F4 — Digital
Digital (137, 153, 6934053)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4153, 6934053, F4, 16) (dual of [6934053, 6933900, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4153, large, F4, 16) (dual of [large, large−153, 17]-code), using
- 9 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 9 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4153, large, F4, 16) (dual of [large, large−153, 17]-code), using
(137, 137+16, large)-Net in Base 4 — Upper bound on s
There is no (137, 153, large)-net in base 4, because
- 14 times m-reduction [i] would yield (137, 139, large)-net in base 4, but