Best Known (218, 218+21, s)-Nets in Base 3
(218, 218+21, 838920)-Net over F3 — Constructive and digital
Digital (218, 239, 838920)-net over F3, using
- 31 times duplication [i] based on digital (217, 238, 838920)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (18, 28, 60)-net over F3, using
- trace code for nets [i] based on digital (4, 14, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- trace code for nets [i] based on digital (4, 14, 30)-net over F9, using
- digital (189, 210, 838860)-net over F3, using
- net defined by OOA [i] based on linear OOA(3210, 838860, F3, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3210, 8388601, F3, 21) (dual of [8388601, 8388391, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3210, 8388601, F3, 21) (dual of [8388601, 8388391, 22]-code), using
- net defined by OOA [i] based on linear OOA(3210, 838860, F3, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
- digital (18, 28, 60)-net over F3, using
- (u, u+v)-construction [i] based on
(218, 218+21, 4194383)-Net over F3 — Digital
Digital (218, 239, 4194383)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3239, 4194383, F3, 2, 21) (dual of [(4194383, 2), 8388527, 22]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(329, 82, F3, 2, 10) (dual of [(82, 2), 135, 11]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(329, 82, F3, 10) (dual of [82, 53, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(329, 96, F3, 10) (dual of [96, 67, 11]-code), using
- construction XX applied to C1 = C({0,1,2,4,26,53}), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,26,53}) [i] based on
- linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,4,26,53}, and minimum distance d ≥ |{−3,−2,…,4}|+1 = 9 (BCH-bound) [i]
- linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(325, 80, F3, 10) (dual of [80, 55, 11]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,4,5,26,53}, and minimum distance d ≥ |{−3,−2,…,6}|+1 = 11 (BCH-bound) [i]
- linear OA(313, 80, F3, 5) (dual of [80, 67, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(33, 11, F3, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(31, 5, F3, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to C1 = C({0,1,2,4,26,53}), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C({0,1,2,4,5,26,53}) [i] based on
- discarding factors / shortening the dual code based on linear OA(329, 96, F3, 10) (dual of [96, 67, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(329, 82, F3, 10) (dual of [82, 53, 11]-code), using
- linear OOA(3210, 4194301, F3, 2, 21) (dual of [(4194301, 2), 8388392, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3210, 8388602, F3, 21) (dual of [8388602, 8388392, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- OOA 2-folding [i] based on linear OA(3210, 8388602, F3, 21) (dual of [8388602, 8388392, 22]-code), using
- linear OOA(329, 82, F3, 2, 10) (dual of [(82, 2), 135, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(218, 218+21, large)-Net in Base 3 — Upper bound on s
There is no (218, 239, large)-net in base 3, because
- 19 times m-reduction [i] would yield (218, 220, large)-net in base 3, but