Organic matter (OM) like compost helps to loosen compacted soil and also resists compaction after it's incorporated. A laboratory soil test would be a good starting point, if you haven't done one in recent years, to both learn how much OM is in the soil and what the soil's other basic characteristics are (primary nutrient levels, acidity level (pH), and if lead or other heavy metals are present to concerning degrees).

If you clear a site for planting vegetables, compost can be mixed into that soil if needed, either by tilling it in with a machine or hand-mixing using a tool like a broadfork, garden fork, or hard-tined rake. Using a biodegradable mulch over exposed soil once the bed is planted not only helps with soil temperature and moisture moderation and suppressing weeds and erosion, but the mulch will self-compost over time and further add OM to the soil below as soil life gradually mixes it in.

A cover crop like daikon radishes does help to loosen soil, feed beneficial soil microbes, and add OM to the soil as it breaks down after dying (or being cut down in preparation for planting). The crop debris can also serve as a short-term mulch as well if left in place. Daikon radish roots are thick and deep, so do help to punch drainage channels into compacted soil. You can learn about some commonly-used cover crops on the linked page, which has a chart that includes planting season information.

Beneficial microbes in the soil produce a natural glue that helps to hold soil particles together, which may sound counter-intuitive as if it will compress the soil further, but it actually improves soil structure that boosts drainage and aeration. While rototilling soil damages soil structure in the short term, it can be an easy way to make an initial compost amendment to get a garden bed started, and the soil structure will recover over time if it isn't tilled regularly.

You can think of the work of the soil microbes like this: they help to turn fine particles into larger clumps, and since larger clumps of soil can't compress as readily as fine particles, there are more air gaps between them that let moisture drain and air reach roots. It's a bit like working with a jar of marbles versus a jar of peppercorns...the spaces between marbles is greater than the peppercorns because they can't pack together as tightly. The same is happening with the soil microbes as they glue tiny soil particles into larger clumps, which gives roots a healthier environment to grow in. OM in the soil fuels beneficial microbe growth, as does having access to live plant roots as year-round as possible, so when the main crop (say, beans and pumpkins) are done for the year, a cover crop can then feed those microbes for at least part of the "off season." Daikon radish is one of the most effective cover crop choices to improve soil compaction because of its substantial root size, but any cover crop can still be helpful with regards to supporting soil health.

Some gardeners are successful converting areas of lawn to garden beds by mowing the grass very short and then covering it with mulch to smother the grass so it dies out. (Wood chips from a tree company are a popular and often low-cost material). Another option is to rent a sod cutter machine to strip the grass off so you have bare soil to work with for amending and then planting and mulching between plants. Competition with other plant roots like turfgrass can stifle the growth of vegetables, so removal of the lawn in areas you're converting (while potentially labor-intensive) is best before new plants are put in. You wouldn't necessarily have to remove the grass below a raised bed you're building. In areas where soil compaction is just too labor-intensive to fix for the short term, you may find that using raised beds is a simpler solution.

Miri

Ok thank you that is very helpful. For the daikon, do I have to till the soil before planting them? I planned to map out a square in my yard and plant only daikon to prep that area for later growing. 

No, you shouldn't need to till the soil before planting the radishes, though having no other competition (like live turfgrass) remaining surrounding the seedlings will help them to establish. Even weakened turf, like having been smothered with a light-blocking tarp (or something similar) for a couple weeks or more, can help the radish seedlings overcome competition from the grass so they can develop and mature.

Miri

Great thank you!

You're welcome!