Best Known (232−25, 232, s)-Nets in Base 3
(232−25, 232, 398583)-Net over F3 — Constructive and digital
Digital (207, 232, 398583)-net over F3, using
- 33 times duplication [i] based on digital (204, 229, 398583)-net over F3, using
- net defined by OOA [i] based on linear OOA(3229, 398583, F3, 25, 25) (dual of [(398583, 25), 9964346, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3229, 4782997, F3, 25) (dual of [4782997, 4782768, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 4783001, F3, 25) (dual of [4783001, 4782772, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [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(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(34, 32, F3, 2) (dual of [32, 28, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 4783001, F3, 25) (dual of [4783001, 4782772, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3229, 4782997, F3, 25) (dual of [4782997, 4782768, 26]-code), using
- net defined by OOA [i] based on linear OOA(3229, 398583, F3, 25, 25) (dual of [(398583, 25), 9964346, 26]-NRT-code), using
(232−25, 232, 1141435)-Net over F3 — Digital
Digital (207, 232, 1141435)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3232, 1141435, F3, 4, 25) (dual of [(1141435, 4), 4565508, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 1195751, F3, 4, 25) (dual of [(1195751, 4), 4782772, 26]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3231, 1195751, F3, 4, 25) (dual of [(1195751, 4), 4782773, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3231, 4783004, F3, 25) (dual of [4783004, 4782773, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3225, 4782970, F3, 25) (dual of [4782970, 4782745, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3197, 4782970, F3, 21) (dual of [4782970, 4782773, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(36, 34, F3, 3) (dual of [34, 28, 4]-code or 34-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 4-folding [i] based on linear OA(3231, 4783004, F3, 25) (dual of [4783004, 4782773, 26]-code), using
- 31 times duplication [i] based on linear OOA(3231, 1195751, F3, 4, 25) (dual of [(1195751, 4), 4782773, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 1195751, F3, 4, 25) (dual of [(1195751, 4), 4782772, 26]-NRT-code), using
(232−25, 232, large)-Net in Base 3 — Upper bound on s
There is no (207, 232, large)-net in base 3, because
- 23 times m-reduction [i] would yield (207, 209, large)-net in base 3, but