What is Gross basis, net basis, implied repo, basis trade?

For a deliverable note with clean price $P$, factor $\mathrm{CF}$ and future price $F$: the gross basis is $P - F\,\mathrm{CF}$; the net basis is the gross basis less the note’s carry to delivery (Proposition 5.3); the implied repo rate is the financing rate at which buying the note, financing it and delivering it into the future exactly breaks even,

with accrued interest $A_0$ today and $A_{\text{del}}$ at delivery, coupons $C$ received in between and $d$ days. A basis trade buys the note and sells the future in the ratio of the factor (long the basis), or the reverse (short the basis).