Best Known (202−23, 202, s)-Nets in Base 4
(202−23, 202, 381315)-Net over F4 — Constructive and digital
Digital (179, 202, 381315)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 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 (165, 188, 381301)-net over F4, using
- net defined by OOA [i] based on linear OOA(4188, 381301, F4, 23, 23) (dual of [(381301, 23), 8769735, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4188, 4194312, F4, 23) (dual of [4194312, 4194124, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4188, 4194315, F4, 23) (dual of [4194315, 4194127, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 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
- discarding factors / shortening the dual code based on linear OA(4188, 4194315, F4, 23) (dual of [4194315, 4194127, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4188, 4194312, F4, 23) (dual of [4194312, 4194124, 24]-code), using
- net defined by OOA [i] based on linear OOA(4188, 381301, F4, 23, 23) (dual of [(381301, 23), 8769735, 24]-NRT-code), using
- digital (3, 14, 14)-net over F4, using
(202−23, 202, 2097187)-Net over F4 — Digital
Digital (179, 202, 2097187)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4202, 2097187, F4, 2, 23) (dual of [(2097187, 2), 4194172, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4202, 4194374, F4, 23) (dual of [4194374, 4194172, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4198, 4194368, F4, 23) (dual of [4194368, 4194170, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(16) [i] based on
- linear OA(4198, 4194370, F4, 20) (dual of [4194370, 4194172, 21]-code), using Gilbert–Varšamov bound and bm = 4198 > Vbs−1(k−1) = 6469 314011 674311 114816 622408 943640 174727 021127 115634 267470 192158 498997 776757 423473 067170 023195 984452 180848 641166 753044 [i]
- linear OA(42, 4, F4, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,4)), using
- Reed–Solomon code RS(2,4) [i]
- linear OA(4198, 4194368, F4, 23) (dual of [4194368, 4194170, 24]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(4202, 4194374, F4, 23) (dual of [4194374, 4194172, 24]-code), using
(202−23, 202, large)-Net in Base 4 — Upper bound on s
There is no (179, 202, large)-net in base 4, because
- 21 times m-reduction [i] would yield (179, 181, large)-net in base 4, but