Best Known (237−21, 237, s)-Nets in Base 4
(237−21, 237, 1677720)-Net over F4 — Constructive and digital
Digital (216, 237, 1677720)-net over F4, using
- 47 times duplication [i] based on digital (209, 230, 1677720)-net over F4, using
- net defined by OOA [i] based on linear OOA(4230, 1677720, F4, 22, 21) (dual of [(1677720, 22), 36909610, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(4230, 8388601, F4, 2, 21) (dual of [(8388601, 2), 16776972, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4230, 8388602, F4, 2, 21) (dual of [(8388602, 2), 16776974, 22]-NRT-code), using
- trace code [i] based on linear OOA(16115, 4194301, F16, 2, 21) (dual of [(4194301, 2), 8388487, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(16115, 8388602, F16, 21) (dual of [8388602, 8388487, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(16115, large, F16, 21) (dual of [large, large−115, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(16115, large, F16, 21) (dual of [large, large−115, 22]-code), using
- OOA 2-folding [i] based on linear OA(16115, 8388602, F16, 21) (dual of [8388602, 8388487, 22]-code), using
- trace code [i] based on linear OOA(16115, 4194301, F16, 2, 21) (dual of [(4194301, 2), 8388487, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4230, 8388602, F4, 2, 21) (dual of [(8388602, 2), 16776974, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(4230, 8388601, F4, 2, 21) (dual of [(8388601, 2), 16776972, 22]-NRT-code), using
- net defined by OOA [i] based on linear OOA(4230, 1677720, F4, 22, 21) (dual of [(1677720, 22), 36909610, 22]-NRT-code), using
(237−21, 237, large)-Net over F4 — Digital
Digital (216, 237, large)-net over F4, using
- t-expansion [i] based on digital (213, 237, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
(237−21, 237, large)-Net in Base 4 — Upper bound on s
There is no (216, 237, large)-net in base 4, because
- 19 times m-reduction [i] would yield (216, 218, large)-net in base 4, but