Best Known (76−17, 76, s)-Nets in Base 5
(76−17, 76, 403)-Net over F5 — Constructive and digital
Digital (59, 76, 403)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (49, 66, 391)-net over F5, using
- net defined by OOA [i] based on linear OOA(566, 391, F5, 17, 17) (dual of [(391, 17), 6581, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(566, 3129, F5, 17) (dual of [3129, 3063, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 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(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(566, 3129, F5, 17) (dual of [3129, 3063, 18]-code), using
- net defined by OOA [i] based on linear OOA(566, 391, F5, 17, 17) (dual of [(391, 17), 6581, 18]-NRT-code), using
- digital (2, 10, 12)-net over F5, using
(76−17, 76, 3682)-Net over F5 — Digital
Digital (59, 76, 3682)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(576, 3682, F5, 17) (dual of [3682, 3606, 18]-code), using
- 542 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 44 times 0, 1, 85 times 0, 1, 147 times 0, 1, 225 times 0) [i] based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 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(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- 542 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 44 times 0, 1, 85 times 0, 1, 147 times 0, 1, 225 times 0) [i] based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
(76−17, 76, 3361035)-Net in Base 5 — Upper bound on s
There is no (59, 76, 3361036)-net in base 5, because
- 1 times m-reduction [i] would yield (59, 75, 3361036)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 26469 784546 817468 066580 043127 851050 830316 879286 841345 > 575 [i]