Best Known (230−27, 230, s)-Nets in Base 4
(230−27, 230, 322642)-Net over F4 — Constructive and digital
Digital (203, 230, 322642)-net over F4, using
- 42 times duplication [i] based on digital (201, 228, 322642)-net over F4, using
- net defined by OOA [i] based on linear OOA(4228, 322642, F4, 27, 27) (dual of [(322642, 27), 8711106, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4228, 4194347, F4, 27) (dual of [4194347, 4194119, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(4228, 4194347, F4, 27) (dual of [4194347, 4194119, 28]-code), using
- net defined by OOA [i] based on linear OOA(4228, 322642, F4, 27, 27) (dual of [(322642, 27), 8711106, 28]-NRT-code), using
(230−27, 230, 1713194)-Net over F4 — Digital
Digital (203, 230, 1713194)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4230, 1713194, F4, 2, 27) (dual of [(1713194, 2), 3426158, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4230, 2097178, F4, 2, 27) (dual of [(2097178, 2), 4194126, 28]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4229, 2097178, F4, 2, 27) (dual of [(2097178, 2), 4194127, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4229, 4194356, F4, 27) (dual of [4194356, 4194127, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(48, 52, F4, 4) (dual of [52, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(4229, 4194356, F4, 27) (dual of [4194356, 4194127, 28]-code), using
- 41 times duplication [i] based on linear OOA(4229, 2097178, F4, 2, 27) (dual of [(2097178, 2), 4194127, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4230, 2097178, F4, 2, 27) (dual of [(2097178, 2), 4194126, 28]-NRT-code), using
(230−27, 230, large)-Net in Base 4 — Upper bound on s
There is no (203, 230, large)-net in base 4, because
- 25 times m-reduction [i] would yield (203, 205, large)-net in base 4, but