Best Known (147−27, 147, s)-Nets in Base 4
(147−27, 147, 1262)-Net over F4 — Constructive and digital
Digital (120, 147, 1262)-net over F4, using
- 41 times duplication [i] based on digital (119, 146, 1262)-net over F4, using
- net defined by OOA [i] based on linear OOA(4146, 1262, F4, 27, 27) (dual of [(1262, 27), 33928, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4146, 16407, F4, 27) (dual of [16407, 16261, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4146, 16410, F4, 27) (dual of [16410, 16264, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4146, 16410, F4, 27) (dual of [16410, 16264, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4146, 16407, F4, 27) (dual of [16407, 16261, 28]-code), using
- net defined by OOA [i] based on linear OOA(4146, 1262, F4, 27, 27) (dual of [(1262, 27), 33928, 28]-NRT-code), using
(147−27, 147, 11111)-Net over F4 — Digital
Digital (120, 147, 11111)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4147, 11111, F4, 27) (dual of [11111, 10964, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4147, 16412, F4, 27) (dual of [16412, 16265, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4147, 16412, F4, 27) (dual of [16412, 16265, 28]-code), using
(147−27, 147, large)-Net in Base 4 — Upper bound on s
There is no (120, 147, large)-net in base 4, because
- 25 times m-reduction [i] would yield (120, 122, large)-net in base 4, but