Strong Singleton type upper bounds for linear insertion-deletion codes

The insertion-delation codes was motivated to correct the synchronization errors. Several new Singleton type upper bounds for the insertion-deletion distances were proved for linear codes over a general finite field ${\bf F}_q$ in 2021. It is a open problem if there exists a sequence of linear $[n_t, k_t]_q$ codes over ${\bf F}_q$ with insdel distances $d_t$ and the lengths $n_t$ go to the infinity, such that $$\lim_{t \longrightarrow \infty} \frac{k_t}{n_t} \geq \frac{1}{2},$$ and $$\lim_{t \longrightarrow \infty} \frac{d_t}{2n_t}>0.$$ In this paper we give some new Singleton type upper bounds for the insertion-deletion distances for linear codes over ${\bf F}_q$, which are much stronger than the previous bounds. Our result implies that asymptotically when the lengths goes to the infinity and the information rate is bigger than or equal to $\frac{1}{2}$ then relative insdel distances go to the zero. More accurately the insdel distances of such linear codes is smaller than a fixed constant. Hence over an arbitrary fixed finite field ${\bf F}_q$ linear codes of rate bigger than or equal to $\frac{1}{2}$ have very small insertion-delation error-correcting capabilities. This partially answers a question proposed by Cheng, Guruswami, Haeupler and Li about the sharp threshold of the rate $\frac{1}{2}$ for linear insertion-deletion codes.