Best Known (59, 59+18, s)-Nets in Base 5
(59, 59+18, 350)-Net over F5 — Constructive and digital
Digital (59, 77, 350)-net over F5, using
- net defined by OOA [i] based on linear OOA(577, 350, F5, 18, 18) (dual of [(350, 18), 6223, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(577, 3150, F5, 18) (dual of [3150, 3073, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(577, 3151, F5, 18) (dual of [3151, 3074, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [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(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(56, 26, F5, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(577, 3151, F5, 18) (dual of [3151, 3074, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(577, 3150, F5, 18) (dual of [3150, 3073, 19]-code), using
(59, 59+18, 3194)-Net over F5 — Digital
Digital (59, 77, 3194)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(577, 3194, F5, 18) (dual of [3194, 3117, 19]-code), using
- 53 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 13 times 0, 1, 30 times 0) [i] based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(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(51, 11, F5, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- 53 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 13 times 0, 1, 30 times 0) [i] based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
(59, 59+18, 990296)-Net in Base 5 — Upper bound on s
There is no (59, 77, 990297)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 661745 090561 362897 692071 954759 964452 928658 341268 853445 > 577 [i]