To become a game programmer you need to take a course on programmingthe best institutes which offer undergraduate course on programming is Computer Science and Engineering offered by Indian institutes of Technology is located at various cities of India if you can qualified JEE advanced in get a seat into IIT for Computer Science and Engineering then I can assure you that you will become a good game programmer because it is a known as the top institute of India in engineering end with the engineering course in Computer Science I can assure you that you can easily learn how to code for games to get into IIT you need to appear for JEE main and then for JEE advanced and get a rank of under 500 to get a seat into IIT for Computer Science and Engineering but remember that that competition is very tough and many aspirants across the country try for IIT for Computer Science and Engineering but very few and up with Computer Science and Engineering at IIT therefore if you are focused to be a game programmer then remain focused and do hard work to achieve your dream.