I agree with the commenters suggesting to sling it across your back. This is my personal method of transport for when I am transporting my 62" longbow. Get or make a 3 point harness that you can place under your arm or at the waist to have control over your case / sling to keep it from moving, and to give you some adjustment on how diagonal it is. I keep my bow nearly upright, just diagonal enough to get past the side of the bicycle. For safety and privacy, I would personally consider fashioning a case of leather, nylon or canvas that simply covered the whole of the katana, and fastened at the top to ensure the sword stayed in its sheath.

I made my holster simply by taking a nylon bow sheath and attaching some nylon straps with buckles to it. Sheath from the archery centre, straps from MEC, easy make. These nylon covers are wide enough to fit recurve bows with thicker pistol grips as well, they would likely hold the katana.

Hm, I suppose I should say that the suggestion to make a sheath bag for your sword is dependant on whether such things will get you in trouble with your regional laws or not. I have personally never been harassed or bothered for transporting a bow in this manner, but bows are covered / encased by standard.