This is my simple implementation of the ARRL Antenna Book design equations for the axial-mode helical antenna. The dimension limits are enforced to ensure axial-mode radiation. Larger C/Λ tightens the radiation pattern at the expense of circularity.
The 50 ohm matching techniques are discussed at length in the ARRL book. I have had good luck with the 1/4-1/2 turn close spaced matching technique, which sacrifices less bandwidth than a coax matching network, but more than building four short helicals in an array with the tapered parallel feed system.