Best Known (157−29, 157, s)-Nets in Base 4
(157−29, 157, 1172)-Net over F4 — Constructive and digital
Digital (128, 157, 1172)-net over F4, using
- 44 times duplication [i] based on digital (124, 153, 1172)-net over F4, using
- net defined by OOA [i] based on linear OOA(4153, 1172, F4, 29, 29) (dual of [(1172, 29), 33835, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4153, 16409, F4, 29) (dual of [16409, 16256, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4153, 16410, F4, 29) (dual of [16410, 16257, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4153, 16410, F4, 29) (dual of [16410, 16257, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4153, 16409, F4, 29) (dual of [16409, 16256, 30]-code), using
- net defined by OOA [i] based on linear OOA(4153, 1172, F4, 29, 29) (dual of [(1172, 29), 33835, 30]-NRT-code), using
(157−29, 157, 10940)-Net over F4 — Digital
Digital (128, 157, 10940)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4157, 10940, F4, 29) (dual of [10940, 10783, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4157, 16417, F4, 29) (dual of [16417, 16260, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4157, 16417, F4, 29) (dual of [16417, 16260, 30]-code), using
(157−29, 157, large)-Net in Base 4 — Upper bound on s
There is no (128, 157, large)-net in base 4, because
- 27 times m-reduction [i] would yield (128, 130, large)-net in base 4, but