Best Known (220, 220+29, s)-Nets in Base 3
(220, 220+29, 113881)-Net over F3 — Constructive and digital
Digital (220, 249, 113881)-net over F3, using
- 31 times duplication [i] based on digital (219, 248, 113881)-net over F3, using
- net defined by OOA [i] based on linear OOA(3248, 113881, F3, 29, 29) (dual of [(113881, 29), 3302301, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3248, 1594335, F3, 29) (dual of [1594335, 1594087, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3248, 1594336, F3, 29) (dual of [1594336, 1594088, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3248, 1594323, F3, 29) (dual of [1594323, 1594075, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3235, 1594323, F3, 28) (dual of [1594323, 1594088, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3248, 1594336, F3, 29) (dual of [1594336, 1594088, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3248, 1594335, F3, 29) (dual of [1594335, 1594087, 30]-code), using
- net defined by OOA [i] based on linear OOA(3248, 113881, F3, 29, 29) (dual of [(113881, 29), 3302301, 30]-NRT-code), using
(220, 220+29, 363846)-Net over F3 — Digital
Digital (220, 249, 363846)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3249, 363846, F3, 4, 29) (dual of [(363846, 4), 1455135, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3249, 398584, F3, 4, 29) (dual of [(398584, 4), 1594087, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3248, 398584, F3, 4, 29) (dual of [(398584, 4), 1594088, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3248, 1594336, F3, 29) (dual of [1594336, 1594088, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3248, 1594323, F3, 29) (dual of [1594323, 1594075, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3235, 1594323, F3, 28) (dual of [1594323, 1594088, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- OOA 4-folding [i] based on linear OA(3248, 1594336, F3, 29) (dual of [1594336, 1594088, 30]-code), using
- 31 times duplication [i] based on linear OOA(3248, 398584, F3, 4, 29) (dual of [(398584, 4), 1594088, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3249, 398584, F3, 4, 29) (dual of [(398584, 4), 1594087, 30]-NRT-code), using
(220, 220+29, large)-Net in Base 3 — Upper bound on s
There is no (220, 249, large)-net in base 3, because
- 27 times m-reduction [i] would yield (220, 222, large)-net in base 3, but