Best Known (165−23, 165, s)-Nets in Base 4
(165−23, 165, 23837)-Net over F4 — Constructive and digital
Digital (142, 165, 23837)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (131, 154, 23832)-net over F4, using
- net defined by OOA [i] based on linear OOA(4154, 23832, F4, 23, 23) (dual of [(23832, 23), 547982, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4154, 262153, F4, 23) (dual of [262153, 261999, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(4154, 262153, F4, 23) (dual of [262153, 261999, 24]-code), using
- net defined by OOA [i] based on linear OOA(4154, 23832, F4, 23, 23) (dual of [(23832, 23), 547982, 24]-NRT-code), using
- digital (0, 11, 5)-net over F4, using
(165−23, 165, 145587)-Net over F4 — Digital
Digital (142, 165, 145587)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4165, 145587, F4, 23) (dual of [145587, 145422, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4165, 262156, F4, 23) (dual of [262156, 261991, 24]-code), using
- (u, u+v)-construction [i] based on
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4165, 262156, F4, 23) (dual of [262156, 261991, 24]-code), using
(165−23, 165, large)-Net in Base 4 — Upper bound on s
There is no (142, 165, large)-net in base 4, because
- 21 times m-reduction [i] would yield (142, 144, large)-net in base 4, but