In this chapter, Moses gives instructions on the cities of refuge.
This chapter begins the "specific stipulations" that relate to the sixth commandment, to not murder. Verses 1-13 detail the cities of refuge, which is a subject that we have seen before. This directly relates to the sixth commandment. Chapter 20 continues with more laws on murder, but there is a brief interlude in verses 14-21 where Moses discusses false witnesses.
This most directly relates to the 9th commandment, to not bear false testimony against your neighbor, but it also implicitly relates to murder because that's one of the things that witnesses would lie about. In that sense, bearing false testimony relates to all of the "social commandments" (the last five, relating to murder, adultery, theft and covetousness) because false testimony is a crime related to the prosecution of the other crimes, just like covetousness is the motivation for these crimes. I would hypothesize that's why this portion is here, but it still doesn't fit into the pattern of specific stipulations mirroring the general stipulations.
The cities of refuge have been discussed at least three times before, and I have written about them on all those occasions. The earliest mention is in Ex 21:13 (my commentary here) when God says that "I will appoint you a place to which he may flee", referring to people who accidentally kill someone and then need to flee from the avenger of blood. As with much of the Exodus account, the passage in Ex 21 is light on detail and we have to look to later chapters to get a better idea of what is going on.
The next mention of cities of refuge is Num 35:6-34 (my commentary here). Num 35 adds substantially more detail and this chapter largely matches the account in Numbers.
Num 35 also says: "If anyone kills a person, the murderer shall be put to death at the evidence of witnesses, but no person shall be put to death on the testimony of one witness". This is a precursor to verses 15-21 in this chapter, which suggests a parallelism. In particular, Numbers discusses witnesses specifically related to murder trials, but this chapter talks about "a single witness... on account of any iniquity or any sin", which means the general category of all witnesses in criminal trials.
Regardless of whether the same person authored both Num 35 and Deut 19, it is very reasonable to suggest they were authored in a similar cultural context, so they were probably written with similar designs or paradigms in mind.
One possibility is that Deut 19 was written to follow the pattern of Num 35 (or vice versa). That is, they are textually correlated. Another possibility is that they have the same author, so the authorship is correlated. A third possibility is that they were written independently, but the authors of these two chapters shared a cultural foundation that expects a discussion of malicious witnesses as part of the discussion of murder. Regardless of which alternative strikes my readers as most plausible, I think that Num 35 as a predecessor of Deut 19 is the best explanation I've seen for why false testimony is discussed here, where it seems out of place.
We can make additional correlations between here and Num 35 to study the current chapter. Probably the strangest part of this chapter is that it talks about adding more cities of refuge. "If the LORD your God enlarges your territory... then you shall add three more cities for yourself, besides these three." This is really strange because in Num 35:6, we were already told there would be six cities of refuge.
Also in Deut 4:41-43 (the third mention of the cities of refuge), Moses had already assigned three cities of refuge east of the Jordan. I think what v. 8-9 in this chapter is talking about is adding the last three cities west of the Jordan when the Israelites invade that land and conquer it. It's just unusual for Moses to use such tentative language, that "if the LORD your God enlarges your territory". A lot of the discussions in Numbers and Deuteronomy are very forward-looking, establishing principles for when they conquer the promised land. There are very few places that talk about if they conquer the promised land, suggesting possible failure.
It's also unusual for the Pentateuch to talk about the land east of the Jordan as their "territory" and the land west of the Jordan as the place they are "enlarged" into. Virtually everything in the Pentateuch focus on the promised land first and Transjordan second, where Transjordan is the place the Israelites are "enlarged" into, and the promised land is their original home.
So I think this chapter is unusual in that regard, but I don't know enough to make this into a bigger point.
I've already written a lot in my prior commentaries about the cities of refuge, so I encourage my readers to review those discussions if they want to learn more. There's very little that's unique about this chapter compared to Num 35, so the prior commentary is still quite relevant to this chapter.
Lastly, v. 14 in this chapter talks about not moving "your neighbor's boundary stone". What that's referring to is boundary markers that would be placed to distinguish between neighboring properties. By moving the marker (presumably in your favor) you are deceptively claiming part of their property as if it were your own. You can buy other properties, but this is essentially talking about theft of land. This passage is probably here because it's considered another kind of deception, like lying witnesses.