Best Known (191, 191+31, s)-Nets in Base 4
(191, 191+31, 17480)-Net over F4 — Constructive and digital
Digital (191, 222, 17480)-net over F4, using
- 42 times duplication [i] based on digital (189, 220, 17480)-net over F4, using
- net defined by OOA [i] based on linear OOA(4220, 17480, F4, 31, 31) (dual of [(17480, 31), 541660, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4220, 262201, F4, 31) (dual of [262201, 261981, 32]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 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 Ce(30) ⊂ Ce(24) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4220, 262201, F4, 31) (dual of [262201, 261981, 32]-code), using
- net defined by OOA [i] based on linear OOA(4220, 17480, F4, 31, 31) (dual of [(17480, 31), 541660, 32]-NRT-code), using
(191, 191+31, 150677)-Net over F4 — Digital
Digital (191, 222, 150677)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4222, 150677, F4, 31) (dual of [150677, 150455, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, 262186, F4, 31) (dual of [262186, 261964, 32]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4181, 262145, F4, 27) (dual of [262145, 261964, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4222, 262186, F4, 31) (dual of [262186, 261964, 32]-code), using
(191, 191+31, large)-Net in Base 4 — Upper bound on s
There is no (191, 222, large)-net in base 4, because
- 29 times m-reduction [i] would yield (191, 193, large)-net in base 4, but