MinT - Best Known (65−10, 65, s)-Nets in Base 32

Best Known (65−10, 65, s)-Nets in Base 32

(65−10, 65, 2202042)-Net over F₃₂ — Constructive and digital

Digital (55, 65, 2202042)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (14, 19, 524322)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹⁹, 524322, F₃₂, 6, 5) (dual of [(524322, 6), 3145913, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(32¹⁹, 524323, F₃₂, 2, 5) (dual of [(524323, 2), 1048627, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(32², 33, F₃₂, 2, 2) (dual of [(33, 2), 64, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;64,32) [i]
        linear OOA(32¹⁷, 524290, F₃₂, 2, 5) (dual of [(524290, 2), 1048563, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(32¹⁷, 1048580, F₃₂, 5) (dual of [1048580, 1048563, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(32¹⁷, 1048576, F₃₂, 5) (dual of [1048576, 1048559, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (36, 46, 1677720)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴⁶, 1677720, F₃₂, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(32⁴⁶, 8388600, F₃₂, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁴⁶, large, F₃₂, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(65−10, 65, 2796200)-Net in Base 32 — Constructive

(55, 65, 2796200)-net in base 32, using

net defined by OOA [i] based on OOA(32⁶⁵, 2796200, S₃₂, 14, 10), using
- OOA 3-folding and stacking with additional row [i] based on OOA(32⁶⁵, 8388601, S₃₂, 2, 10), using
  - discarding factors based on OOA(32⁶⁵, 8388602, S₃₂, 2, 10), using
    - discarding parts of the base [i] based on linear OOA(64⁵⁴, 8388602, F₆₄, 2, 10) (dual of [(8388602, 2), 16777150, 11]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(64¹⁷, 4194301, F₆₄, 2, 5) (dual of [(4194301, 2), 8388585, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64¹⁷, 8388602, F₆₄, 5) (dual of [8388602, 8388585, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹⁷, large, F₆₄, 5) (dual of [large, large−17, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OOA(64³⁷, 4194301, F₆₄, 2, 10) (dual of [(4194301, 2), 8388565, 11]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64³⁷, 8388602, F₆₄, 10) (dual of [8388602, 8388565, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(65−10, 65, large)-Net over F₃₂ — Digital

Digital (55, 65, large)-net over F₃₂, using

t-expansion [i] based on digital (52, 65, large)-net over F₃₂, using
- 1 times m-reduction [i] based on digital (52, 66, large)-net over F₃₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁶⁶, large, F₃₂, 14) (dual of [large, large−66, 15]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(65−10, 65, large)-Net in Base 32 — Upper bound on s

There is no (55, 65, large)-net in base 32, because

8 times m-reduction [i] would yield (55, 57, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (65−10, 65, s)-Nets in Base 32

(65−10, 65, 2202042)-Net over F32 — Constructive and digital

(65−10, 65, 2796200)-Net in Base 32 — Constructive

(65−10, 65, large)-Net over F32 — Digital

(65−10, 65, large)-Net in Base 32 — Upper bound on s

(65−10, 65, 2202042)-Net over F₃₂ — Constructive and digital

(65−10, 65, large)-Net over F₃₂ — Digital