Best Known (215−24, 215, s)-Nets in Base 4
(215−24, 215, 349542)-Net over F4 — Constructive and digital
Digital (191, 215, 349542)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 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 (174, 198, 349525)-net over F4, using
- net defined by OOA [i] based on linear OOA(4198, 349525, F4, 24, 24) (dual of [(349525, 24), 8388402, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4198, 4194300, F4, 24) (dual of [4194300, 4194102, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4198, 4194304, F4, 24) (dual of [4194304, 4194106, 25]-code), using
- 1 times truncation [i] based on linear OA(4199, 4194305, F4, 25) (dual of [4194305, 4194106, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4199, 4194305, F4, 25) (dual of [4194305, 4194106, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4198, 4194304, F4, 24) (dual of [4194304, 4194106, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4198, 4194300, F4, 24) (dual of [4194300, 4194102, 25]-code), using
- net defined by OOA [i] based on linear OOA(4198, 349525, F4, 24, 24) (dual of [(349525, 24), 8388402, 25]-NRT-code), using
- digital (5, 17, 17)-net over F4, using
(215−24, 215, 2168353)-Net over F4 — Digital
Digital (191, 215, 2168353)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4215, 2168353, F4, 24) (dual of [2168353, 2168138, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4215, 4194387, F4, 24) (dual of [4194387, 4194172, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4199, 4194305, F4, 25) (dual of [4194305, 4194106, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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]
- linear OA(416, 82, F4, 6) (dual of [82, 66, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(416, 85, F4, 6) (dual of [85, 69, 7]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4215, 4194387, F4, 24) (dual of [4194387, 4194172, 25]-code), using
(215−24, 215, large)-Net in Base 4 — Upper bound on s
There is no (191, 215, large)-net in base 4, because
- 22 times m-reduction [i] would yield (191, 193, large)-net in base 4, but