Best Known (63, 63+18, s)-Nets in Base 5
(63, 63+18, 357)-Net over F5 — Constructive and digital
Digital (63, 81, 357)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- a shift-net [i]
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (53, 71, 347)-net over F5, using
- net defined by OOA [i] based on linear OOA(571, 347, F5, 18, 18) (dual of [(347, 18), 6175, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(571, 3123, F5, 18) (dual of [3123, 3052, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(571, 3123, F5, 18) (dual of [3123, 3052, 19]-code), using
- net defined by OOA [i] based on linear OOA(571, 347, F5, 18, 18) (dual of [(347, 18), 6175, 19]-NRT-code), using
- digital (1, 10, 10)-net over F5, using
(63, 63+18, 3880)-Net over F5 — Digital
Digital (63, 81, 3880)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(581, 3880, F5, 18) (dual of [3880, 3799, 19]-code), using
- 740 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 5 times 0, 1, 13 times 0, 1, 30 times 0, 1, 64 times 0, 1, 123 times 0, 1, 205 times 0, 1, 290 times 0) [i] based on linear OA(571, 3130, F5, 18) (dual of [3130, 3059, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 740 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 5 times 0, 1, 13 times 0, 1, 30 times 0, 1, 64 times 0, 1, 123 times 0, 1, 205 times 0, 1, 290 times 0) [i] based on linear OA(571, 3130, F5, 18) (dual of [3130, 3059, 19]-code), using
(63, 63+18, 2024977)-Net in Base 5 — Upper bound on s
There is no (63, 81, 2024978)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 413 591625 315782 908538 614277 762122 453445 328847 040446 442825 > 581 [i]