Best Known (209−27, 209, s)-Nets in Base 4
(209−27, 209, 80663)-Net over F4 — Constructive and digital
Digital (182, 209, 80663)-net over F4, using
- net defined by OOA [i] based on linear OOA(4209, 80663, F4, 27, 27) (dual of [(80663, 27), 2177692, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4209, 1048620, F4, 27) (dual of [1048620, 1048411, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 1048624, F4, 27) (dual of [1048624, 1048415, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 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
- discarding factors / shortening the dual code based on linear OA(4209, 1048624, F4, 27) (dual of [1048624, 1048415, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4209, 1048620, F4, 27) (dual of [1048620, 1048411, 28]-code), using
(209−27, 209, 509322)-Net over F4 — Digital
Digital (182, 209, 509322)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4209, 509322, F4, 2, 27) (dual of [(509322, 2), 1018435, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4209, 524312, F4, 2, 27) (dual of [(524312, 2), 1048415, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4209, 1048624, F4, 27) (dual of [1048624, 1048415, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 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(4209, 1048624, F4, 27) (dual of [1048624, 1048415, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(4209, 524312, F4, 2, 27) (dual of [(524312, 2), 1048415, 28]-NRT-code), using
(209−27, 209, large)-Net in Base 4 — Upper bound on s
There is no (182, 209, large)-net in base 4, because
- 25 times m-reduction [i] would yield (182, 184, large)-net in base 4, but