Best Known (126, 126+17, s)-Nets in Base 4
(126, 126+17, 524298)-Net over F4 — Constructive and digital
Digital (126, 143, 524298)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- a shift-net [i]
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (116, 133, 524288)-net over F4, using
- net defined by OOA [i] based on linear OOA(4133, 524288, F4, 17, 17) (dual of [(524288, 17), 8912763, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [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]
- OOA 8-folding and stacking with additional row [i] based on linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using
- net defined by OOA [i] based on linear OOA(4133, 524288, F4, 17, 17) (dual of [(524288, 17), 8912763, 18]-NRT-code), using
- digital (2, 10, 10)-net over F4, using
(126, 126+17, 2097179)-Net over F4 — Digital
Digital (126, 143, 2097179)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4143, 2097179, F4, 2, 17) (dual of [(2097179, 2), 4194215, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4143, 4194358, F4, 17) (dual of [4194358, 4194215, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4143, 4194359, F4, 17) (dual of [4194359, 4194216, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [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]
- linear OA(489, 4194305, F4, 11) (dual of [4194305, 4194216, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(410, 54, F4, 5) (dual of [54, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4143, 4194359, F4, 17) (dual of [4194359, 4194216, 18]-code), using
- OOA 2-folding [i] based on linear OA(4143, 4194358, F4, 17) (dual of [4194358, 4194215, 18]-code), using
(126, 126+17, large)-Net in Base 4 — Upper bound on s
There is no (126, 143, large)-net in base 4, because
- 15 times m-reduction [i] would yield (126, 128, large)-net in base 4, but