Best Known (251−36, 251, s)-Nets in Base 4
(251−36, 251, 14565)-Net over F4 — Constructive and digital
Digital (215, 251, 14565)-net over F4, using
- 42 times duplication [i] based on digital (213, 249, 14565)-net over F4, using
- t-expansion [i] based on digital (212, 249, 14565)-net over F4, using
- net defined by OOA [i] based on linear OOA(4249, 14565, F4, 37, 37) (dual of [(14565, 37), 538656, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4249, 262171, F4, 37) (dual of [262171, 261922, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4249, 262176, F4, 37) (dual of [262176, 261927, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(32) [i] based on
- linear OA(4244, 262144, F4, 37) (dual of [262144, 261900, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4217, 262144, F4, 33) (dual of [262144, 261927, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-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(36) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(4249, 262176, F4, 37) (dual of [262176, 261927, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4249, 262171, F4, 37) (dual of [262171, 261922, 38]-code), using
- net defined by OOA [i] based on linear OOA(4249, 14565, F4, 37, 37) (dual of [(14565, 37), 538656, 38]-NRT-code), using
- t-expansion [i] based on digital (212, 249, 14565)-net over F4, using
(251−36, 251, 131093)-Net over F4 — Digital
Digital (215, 251, 131093)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4251, 131093, F4, 2, 36) (dual of [(131093, 2), 261935, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4251, 262186, F4, 36) (dual of [262186, 261935, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4251, 262187, F4, 36) (dual of [262187, 261936, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- linear OA(4244, 262144, F4, 37) (dual of [262144, 261900, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 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(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4251, 262187, F4, 36) (dual of [262187, 261936, 37]-code), using
- OOA 2-folding [i] based on linear OA(4251, 262186, F4, 36) (dual of [262186, 261935, 37]-code), using
(251−36, 251, large)-Net in Base 4 — Upper bound on s
There is no (215, 251, large)-net in base 4, because
- 34 times m-reduction [i] would yield (215, 217, large)-net in base 4, but