Best Known (213, 213+37, s)-Nets in Base 4
(213, 213+37, 14565)-Net over F4 — Constructive and digital
Digital (213, 250, 14565)-net over F4, using
- 41 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
(213, 213+37, 111108)-Net over F4 — Digital
Digital (213, 250, 111108)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4250, 111108, F4, 2, 37) (dual of [(111108, 2), 221966, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4250, 131088, F4, 2, 37) (dual of [(131088, 2), 261926, 38]-NRT-code), using
- 41 times duplication [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
- 41 times duplication [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(4250, 131088, F4, 2, 37) (dual of [(131088, 2), 261926, 38]-NRT-code), using
(213, 213+37, large)-Net in Base 4 — Upper bound on s
There is no (213, 250, large)-net in base 4, because
- 35 times m-reduction [i] would yield (213, 215, large)-net in base 4, but