Best Known (252−31, 252, s)-Nets in Base 4
(252−31, 252, 69923)-Net over F4 — Constructive and digital
Digital (221, 252, 69923)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 20, 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 (201, 232, 69906)-net over F4, using
- net defined by OOA [i] based on linear OOA(4232, 69906, F4, 31, 31) (dual of [(69906, 31), 2166854, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4232, 1048591, F4, 31) (dual of [1048591, 1048359, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4232, 1048597, F4, 31) (dual of [1048597, 1048365, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4232, 1048597, F4, 31) (dual of [1048597, 1048365, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4232, 1048591, F4, 31) (dual of [1048591, 1048359, 32]-code), using
- net defined by OOA [i] based on linear OOA(4232, 69906, F4, 31, 31) (dual of [(69906, 31), 2166854, 32]-NRT-code), using
- digital (5, 20, 17)-net over F4, using
(252−31, 252, 632292)-Net over F4 — Digital
Digital (221, 252, 632292)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4252, 632292, F4, 31) (dual of [632292, 632040, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4252, 1048598, F4, 31) (dual of [1048598, 1048346, 32]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(410, 11, F4, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,4)), using
- dual of repetition code with length 11 [i]
- linear OA(411, 11, F4, 11) (dual of [11, 0, 12]-code or 11-arc in PG(10,4)), using
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(410, 11, F4, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,4)), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4252, 1048598, F4, 31) (dual of [1048598, 1048346, 32]-code), using
(252−31, 252, large)-Net in Base 4 — Upper bound on s
There is no (221, 252, large)-net in base 4, because
- 29 times m-reduction [i] would yield (221, 223, large)-net in base 4, but