Best Known (253−37, 253, s)-Nets in Base 4
(253−37, 253, 14565)-Net over F4 — Constructive and digital
Digital (216, 253, 14565)-net over F4, using
- 44 times duplication [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
(253−37, 253, 125568)-Net over F4 — Digital
Digital (216, 253, 125568)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4253, 125568, F4, 2, 37) (dual of [(125568, 2), 250883, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4253, 131090, F4, 2, 37) (dual of [(131090, 2), 261927, 38]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4249, 131088, F4, 2, 37) (dual of [(131088, 2), 261927, 38]-NRT-code), using
- OOA 2-folding [i] 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
- OOA 2-folding [i] based on linear OA(4249, 262176, F4, 37) (dual of [262176, 261927, 38]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4249, 131088, F4, 2, 37) (dual of [(131088, 2), 261927, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4253, 131090, F4, 2, 37) (dual of [(131090, 2), 261927, 38]-NRT-code), using
(253−37, 253, large)-Net in Base 4 — Upper bound on s
There is no (216, 253, large)-net in base 4, because
- 35 times m-reduction [i] would yield (216, 218, large)-net in base 4, but