Best Known (234−25, 234, s)-Nets in Base 3
(234−25, 234, 398584)-Net over F3 — Constructive and digital
Digital (209, 234, 398584)-net over F3, using
- 31 times duplication [i] based on digital (208, 233, 398584)-net over F3, using
- net defined by OOA [i] based on linear OOA(3233, 398584, F3, 25, 25) (dual of [(398584, 25), 9964367, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3233, 4783009, F3, 25) (dual of [4783009, 4782776, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, 4783010, F3, 25) (dual of [4783010, 4782777, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3225, 4782969, F3, 25) (dual of [4782969, 4782744, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3233, 4783010, F3, 25) (dual of [4783010, 4782777, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3233, 4783009, F3, 25) (dual of [4783009, 4782776, 26]-code), using
- net defined by OOA [i] based on linear OOA(3233, 398584, F3, 25, 25) (dual of [(398584, 25), 9964367, 26]-NRT-code), using
(234−25, 234, 1195755)-Net over F3 — Digital
Digital (209, 234, 1195755)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3234, 1195755, F3, 4, 25) (dual of [(1195755, 4), 4782786, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3234, 4783020, F3, 25) (dual of [4783020, 4782786, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3225, 4782969, F3, 25) (dual of [4782969, 4782744, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(39, 51, F3, 4) (dual of [51, 42, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- OOA 4-folding [i] based on linear OA(3234, 4783020, F3, 25) (dual of [4783020, 4782786, 26]-code), using
(234−25, 234, large)-Net in Base 3 — Upper bound on s
There is no (209, 234, large)-net in base 3, because
- 23 times m-reduction [i] would yield (209, 211, large)-net in base 3, but