Best Known (200, 239, s)-Nets in Base 4
(200, 239, 3450)-Net over F4 — Constructive and digital
Digital (200, 239, 3450)-net over F4, using
- 45 times duplication [i] based on digital (195, 234, 3450)-net over F4, using
- net defined by OOA [i] based on linear OOA(4234, 3450, F4, 39, 39) (dual of [(3450, 39), 134316, 40]-NRT-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4234, 65551, F4, 39) (dual of [65551, 65317, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4234, 65553, F4, 39) (dual of [65553, 65319, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(36) [i] based on
- linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(38) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(4234, 65553, F4, 39) (dual of [65553, 65319, 40]-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4234, 65551, F4, 39) (dual of [65551, 65317, 40]-code), using
- net defined by OOA [i] based on linear OOA(4234, 3450, F4, 39, 39) (dual of [(3450, 39), 134316, 40]-NRT-code), using
(200, 239, 36404)-Net over F4 — Digital
Digital (200, 239, 36404)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4239, 36404, F4, 39) (dual of [36404, 36165, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4239, 65567, F4, 39) (dual of [65567, 65328, 40]-code), using
- construction XX applied to Ce(38) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-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(38) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4239, 65567, F4, 39) (dual of [65567, 65328, 40]-code), using
(200, 239, large)-Net in Base 4 — Upper bound on s
There is no (200, 239, large)-net in base 4, because
- 37 times m-reduction [i] would yield (200, 202, large)-net in base 4, but