Best Known (131−25, 131, s)-Nets in Base 3
(131−25, 131, 688)-Net over F3 — Constructive and digital
Digital (106, 131, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (106, 132, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 33, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 33, 172)-net over F81, using
(131−25, 131, 2820)-Net over F3 — Digital
Digital (106, 131, 2820)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3131, 2820, F3, 2, 25) (dual of [(2820, 2), 5509, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3131, 3286, F3, 2, 25) (dual of [(3286, 2), 6441, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3131, 6572, F3, 25) (dual of [6572, 6441, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3131, 6572, F3, 25) (dual of [6572, 6441, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(3131, 3286, F3, 2, 25) (dual of [(3286, 2), 6441, 26]-NRT-code), using
(131−25, 131, 390060)-Net in Base 3 — Upper bound on s
There is no (106, 131, 390061)-net in base 3, because
- 1 times m-reduction [i] would yield (106, 130, 390061)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 106 112607 614702 737203 461828 591742 196038 010090 926710 370386 153073 > 3130 [i]