Best Known (197, 197+26, s)-Nets in Base 3
(197, 197+26, 122641)-Net over F3 — Constructive and digital
Digital (197, 223, 122641)-net over F3, using
- 31 times duplication [i] based on digital (196, 222, 122641)-net over F3, using
- net defined by OOA [i] based on linear OOA(3222, 122641, F3, 26, 26) (dual of [(122641, 26), 3188444, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3222, 1594333, F3, 26) (dual of [1594333, 1594111, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 1594336, F3, 26) (dual of [1594336, 1594114, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3222, 1594323, F3, 26) (dual of [1594323, 1594101, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 1594336, F3, 26) (dual of [1594336, 1594114, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3222, 1594333, F3, 26) (dual of [1594333, 1594111, 27]-code), using
- net defined by OOA [i] based on linear OOA(3222, 122641, F3, 26, 26) (dual of [(122641, 26), 3188444, 27]-NRT-code), using
(197, 197+26, 398584)-Net over F3 — Digital
Digital (197, 223, 398584)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3223, 398584, F3, 4, 26) (dual of [(398584, 4), 1594113, 27]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3222, 398584, F3, 4, 26) (dual of [(398584, 4), 1594114, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3222, 1594336, F3, 26) (dual of [1594336, 1594114, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3222, 1594323, F3, 26) (dual of [1594323, 1594101, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- OOA 4-folding [i] based on linear OA(3222, 1594336, F3, 26) (dual of [1594336, 1594114, 27]-code), using
- 31 times duplication [i] based on linear OOA(3222, 398584, F3, 4, 26) (dual of [(398584, 4), 1594114, 27]-NRT-code), using
(197, 197+26, large)-Net in Base 3 — Upper bound on s
There is no (197, 223, large)-net in base 3, because
- 24 times m-reduction [i] would yield (197, 199, large)-net in base 3, but