AN ACR FLT MAKES A CAT IIIB ILS APCH INTO LAX; RWY 25L AND TURNS ONTO A CLOSED TXWY K AT LAX CA.

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PERFORMED ACTUAL AUTOLAND CAT IIIB APCH RWY 25L LAX. RVR TOUCHDOWN 1000 FT; MID 1000 FT; ROLLOUT 800 FT. DURING ROLLOUT; ATC DIRECTED US TO MAKE R TURN OFF OF TXWY M. THAT WAS OUR PLAN AS WE UNDERSTOOD TXWY K WAS CLOSED. DURING DECELERATION; WE SAW THE GREEN TURNOFF LIGHTS (TAXI) ASSUMING IT WAS TXWY M BECAUSE WE THOUGHT WE PASSED TXWY K; WE MADE THE R TURN. BY THE TIME WE WERE 1/2 WAY ONTO THE TXWY; WE REALIZED IT WAS INDEED TXWY K. TWR TOLD US WE WERE ON TXWY K AND HAD FOLLOW-ME TRUCK CLR THE TXWY SO THAT WE COULD PROCEED TO THE GATE. IN REALITY; TXWY K WAS NOT CLOSED (IT WAS LIT FOR TAXI AND THERE WERE NO BARRICADES OR OBSTRUCTIONS). IT WAS SO FOGGY THAT WE ASSUMED TXWY K WAS TXWY M. THERE WERE NO PROBS WITH TWR. NO DELAYS -- JUST A MISUNDERSTANDING ON THE TURNOFF. IF TXWY K WAS INDEED CLOSED; THE TURNOFF TAXI (GREEN) LIGHTS SHOULD HAVE BEEN TURNED OFF WITH A FORM OF LIGHTED BARRICADE TO ENHANCE THE FACT THAT IT WAS CLOSED. WE; IN THE FOG DECELERATING ON ROLLOUT; MISTOOK THE WELL LIT TXWY FOR TXWY M. EVEN THE SIGNAGE WAS HARD TO SEE.