Best Known (127, 127+16, s)-Nets in Base 4
(127, 127+16, 524302)-Net over F4 — Constructive and digital
Digital (127, 143, 524302)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 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 (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 (3, 11, 14)-net over F4, using
(127, 127+16, 2575984)-Net over F4 — Digital
Digital (127, 143, 2575984)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4143, 2575984, F4, 16) (dual of [2575984, 2575841, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4143, 4194368, F4, 16) (dual of [4194368, 4194225, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(478, 4194304, F4, 10) (dual of [4194304, 4194226, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4143, 4194368, F4, 16) (dual of [4194368, 4194225, 17]-code), using
(127, 127+16, large)-Net in Base 4 — Upper bound on s
There is no (127, 143, large)-net in base 4, because
- 14 times m-reduction [i] would yield (127, 129, large)-net in base 4, but