What is difference between 'antenna rating' and 'vs L3 DSN'?

[CHANDO]

For example communotron16 has 500k antenna rating. that 'k' is abbreviation of 'km' right?
I thought it means it can communicate with object 500km away.
But below 'antenna rating' there is 'vs L3 DSN 354Mm'
354Mm is abbreviation of 354million meter, in other word 354000km.
Then is this means communotron 16 can communicate object with 354000km away?

500km and 354000km which one is right?

[Markus Reese]

I am still learning it a bit, but I believe the L is the rating of the tracking station. So the 500km is say if you had a mothership in orbit to communicate with a probe.

That L3 is the range to transmit with kerbin if you have the level 3 tracking station.

Additionally is the direct vs relay. I am trying to figurd thst out, but it seems that a direct can only communicate line of sight while a relat can receive and transmit? Got stumped putting up my satellite network because my ship which was direct couldnt seem to relay through the network of sats I deployed.

[Jupiter3927]

Antenna rating is how far your antennas are able to communicate.
Two 500km rated antennas can only communicate from at most 500km away.
Your level 3 tracking station should have a rating of 150Gm or 250,000,000km.
Antennas take the square root of your antennas multiplied to determine maximum range so your wimpy antenna on your ship with 500km paired with your overpowered tracking station gets a maximum range of (500*250,000,000)^(1/2) or 353,553km.

Your communications will take the shortest possible route to send data home so they will take a direct connection to KSC if possible.
Relays will be used if KSC isn't in your line of sight like if it's blocked by a planet or if it's too far away.

There's a handy wiki page here...
https://wiki.kerbalspaceprogram.com/wiki/CommNet
...that tells you everything you need to know about relays and more.