Best Known (178−23, 178, s)-Nets in Base 4
(178−23, 178, 95328)-Net over F4 — Constructive and digital
Digital (155, 178, 95328)-net over F4, using
- 42 times duplication [i] based on digital (153, 176, 95328)-net over F4, using
- net defined by OOA [i] based on linear OOA(4176, 95328, F4, 23, 23) (dual of [(95328, 23), 2192368, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4176, 1048609, F4, 23) (dual of [1048609, 1048433, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4176, 1048611, F4, 23) (dual of [1048611, 1048435, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(4176, 1048611, F4, 23) (dual of [1048611, 1048435, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4176, 1048609, F4, 23) (dual of [1048609, 1048433, 24]-code), using
- net defined by OOA [i] based on linear OOA(4176, 95328, F4, 23, 23) (dual of [(95328, 23), 2192368, 24]-NRT-code), using
(178−23, 178, 524309)-Net over F4 — Digital
Digital (155, 178, 524309)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4178, 524309, F4, 2, 23) (dual of [(524309, 2), 1048440, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4178, 1048618, F4, 23) (dual of [1048618, 1048440, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4178, 1048619, F4, 23) (dual of [1048619, 1048441, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4178, 1048619, F4, 23) (dual of [1048619, 1048441, 24]-code), using
- OOA 2-folding [i] based on linear OA(4178, 1048618, F4, 23) (dual of [1048618, 1048440, 24]-code), using
(178−23, 178, large)-Net in Base 4 — Upper bound on s
There is no (155, 178, large)-net in base 4, because
- 21 times m-reduction [i] would yield (155, 157, large)-net in base 4, but