Best Known (114, 114+36, s)-Nets in Base 3
(114, 114+36, 400)-Net over F3 — Constructive and digital
Digital (114, 150, 400)-net over F3, using
- 32 times duplication [i] based on digital (112, 148, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 37, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 37, 100)-net over F81, using
(114, 114+36, 796)-Net over F3 — Digital
Digital (114, 150, 796)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3150, 796, F3, 36) (dual of [796, 646, 37]-code), using
- 42 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0) [i] based on linear OA(3144, 748, F3, 36) (dual of [748, 604, 37]-code), using
- construction XX applied to C1 = C([725,31]), C2 = C([0,33]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([725,33]) [i] based on
- linear OA(3136, 728, F3, 35) (dual of [728, 592, 36]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,31}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3130, 728, F3, 34) (dual of [728, 598, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3142, 728, F3, 37) (dual of [728, 586, 38]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,33}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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 (see above)
- construction XX applied to C1 = C([725,31]), C2 = C([0,33]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([725,33]) [i] based on
- 42 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0) [i] based on linear OA(3144, 748, F3, 36) (dual of [748, 604, 37]-code), using
(114, 114+36, 35718)-Net in Base 3 — Upper bound on s
There is no (114, 150, 35719)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 370010 801554 298112 274052 876648 973752 575352 722183 902757 439193 313579 912045 > 3150 [i]